Weights on random operators and formal dimension

Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revisions are self-checked by the writing AI. The October revision also corrects points found by GPT-6 Astra (OpenAI), Ultra, in a separate review session. The proofs in (B4) and (B9) were drafted by GPT-6 Astra (OpenAI) in ChatGPT web, Pro mode, and checked and adapted by Claude Opus 5.5, October 2026. Public domain (CC0).

Introduction

On a measure space, a positive function \(\rho\) turns into a measure \(\rho\,d\mu\), and integrating the constant function \(1\) against it gives a number. This lesson does the same thing on the "space of orbits" of a measured groupoid, where functions are replaced by random operators. The earlier lessons of this course built the pieces: a measurable groupoid \(G\), a transverse measure \(\Lambda\) of modulus \(\delta\), square-integrable representations (random Hilbert spaces) \(H\), and the von Neumann algebra \(\operatorname{End}_\Lambda(H)\) of random operators. Here we attach to every positive random operator \(T\) of degree one a normal semifinite weight \(\varphi_T\) on \(\operatorname{End}_\Lambda(H)\). Heuristically \[ \varphi_T(A)=\int\operatorname{Trace}(T_xA_x)\,d\Lambda(x), \] an integral over the orbit space of a function that is only defined up to the transformation rule of \(T\). The weight is characterized by its values on the "continuous sums of rank-one operators" \(\theta_\nu(\xi,\xi)\), and every normal semifinite weight arises this way (Theorem 5.3).

The main results are the following.

1. The map \(T\mapsto\varphi_T\) is a bijection from positive random operators of degree one onto normal semifinite weights on \(\operatorname{End}_\Lambda(H)\). For nonsingular \(T\), which give the faithful weights, the modular group of \(\varphi_T\) is \(\operatorname{Ad}T_x^{it}\), and cocycle derivatives are quotients \(T_{2,x}^{it}T_{1,x}^{-it}\) (Section 5).
2. An operator-valued weight \(E_\nu\) from the direct integral of the algebras \(\mathcal L(H_x)\) onto \(\operatorname{End}_\Lambda(H)\) is the average along orbits, and \(\varphi_T\circ E_\nu\) is the weight \(\int\operatorname{Trace}(T_x\,\cdot\,)\,d\Lambda_\nu\) (Section 6).
3. When the isotropy groups are trivial and \(T\) is nonsingular, the modular spectrum of \(\varphi_T\) is read off from the spectra of the operators \(\log T_x\), and \(\varphi_T\) is a trace exactly when \(T_x\) is scalar (Section 7).
4. For nonsingular \(T\), the modular group of \(\varphi_T\) is integrable exactly when the \(T_x\) have absolutely continuous spectrum (for diagonal \(T\)); its centralizer is then an algebra of random operators on the stable kernel of \(\delta\) (Section 8).
5. Integration formulas: \(\varphi_T(1)=\int F_T\,d\Lambda\) for an explicit random variable \(F_T\), a Hilbert–Schmidt formula for convolution operators, and invariance under proper homomorphisms (Section 9).
6. When \(\delta=1\), the formal dimension \(\dim_\Lambda H=\varphi_1(1)\) of a random Hilbert space: additivity, invariance, agreement with the formal degree and with the Murray–von Neumann dimension for unimodular groups, and the value \(\Lambda(\nu_T)\) for a transversal (Section 10).
7. The index \(\operatorname{Ind}_\Lambda\) of a random Fredholm operator, a real number that is additive, locally constant and invariant under \(\Lambda\)-compact perturbations (Section 11).

Items 6 and 7 are what make the theory useful for index theory on foliations: the \(\Lambda\)-index of a leafwise elliptic operator is a real number of exactly this kind.

What is assumed. The lesson relies on three earlier lessons of this course: "The spatial derivative", "Measured groupoids and transverse measures" and "Square-integrable representations and random operators". Section 1 restates precisely what is used from them. It also uses Tomita–Takesaki theory, normal weights and the Connes cocycle derivative, operator-valued weights, direct integrals, and the Arveson spectrum of a one-parameter automorphism group. These are listed in Section 2, "Results used from other lessons", with the place where each is proved; the Arveson spectrum (B4) and the facts on unimodular groups (B9) are proved there, from the lessons on harmonic analysis.

Basic references are [Connes 1979], [Connes 1980a], [Haagerup 1979] and [Connes 1994].

Conventions.

• Inner products are linear in the first variable. For a vector \(\xi\), \(\xi\xi^*\) is the rank-one operator \(\zeta\mapsto\langle\zeta,\xi\rangle\xi\). Hilbert spaces and measurable fields of Hilbert spaces are separable.
• Let \(T\) be positive and self-adjoint. For a vector \(\zeta\) we write \(\langle T\zeta,\zeta\rangle= \|T^{1/2}\zeta\|^2\) if \(\zeta\in\operatorname{dom}T^{1/2}\) and \(\langle T\zeta,\zeta\rangle=+\infty\) otherwise. \(T^{it}\) is defined by functional calculus with \(0^{it}=0\): a partial isometry with initial and final projection the support \(s(T)\), the projection onto \((\ker T)^\perp\). \(T\) is nonsingular if \(s(T)=1\).
• Weights are additive positively homogeneous maps \(A_+\to[0,\infty]\); normal means \(\varphi(\sup x_i)= \sup\varphi(x_i)\) for bounded increasing nets; fns abbreviates faithful normal semifinite. \(\sigma^\varphi\) is the modular group, \((D\varphi_2:D\varphi_1)_t\) the Connes cocycle derivative, \(s(\varphi)\) the support. A normal weight extends by normality to the extended positive part \(\widehat A_+\) (increasing limits of elements of \(A_+\)).
• Standing assumptions. \(G\) is a measurable groupoid whose Borel structure is standard, and \(G\) has a faithful proper transverse function (hypothesis (F) of the earlier lessons). \(\delta:G\to\mathbb R_+^*\) is a Borel homomorphism, and \(\Lambda\) is a \(\sigma\)-finite transverse measure of modulus \(\delta\) ("module \(\delta\)" in the lesson "Measured groupoids and transverse measures"). Then every measure \(\Lambda_\nu\) is \(\sigma\)-finite, every algebra \(\operatorname{End}_\Lambda(H)\) is a von Neumann algebra with separable predual, and in particular it has a faithful normal state. The lesson "Square-integrable representations and random operators" constructs \(\operatorname{End}_\Lambda(H)\) only for \(\sigma\)-finite \(\Lambda\), and Example 6.13 of "Measured groupoids and transverse measures" shows that semi-finite transverse measures can behave differently; so everything below is stated for \(\sigma\)-finite \(\Lambda\).

1. What we use from the earlier lessons

1.1 Transverse functions and transverse measures

The following is taught in "Measured groupoids and transverse measures"; the numbers in parentheses after each item refer to that lesson.

(R1) Kernels and transverse functions. A kernel \(\lambda\) on \(G\) assigns to every \(y\in G^{(0)}\) a positive measure \(\lambda^y\) on \(G^y=r^{-1}(y)\), measurably in \(y\) (and s-finitely). It is proper if \(G\) is an increasing union of Borel sets \(A_n\) with \(\gamma\mapsto\lambda^{s(\gamma)}(\gamma^{-1}A_n)\) bounded. A transverse function is a kernel \(\nu\) with \(\gamma\nu^x=\nu^y\) for all \(\gamma:x\to y\) (left invariance); \(\mathcal E^+\) is the cone of proper transverse functions, stable under sums. \(\nu\) is faithful if \(\nu^y\neq0\) for all \(y\). For \(\nu\in\mathcal E^+\) and a finite-valued Borel \(g\ge0\) on \(G^{(0)}\), \((g\circ s)\nu\in\mathcal E^+\). For \(f\ge0\) on \(G\) we write \(\nu(f)(y)=\int f\,d\nu^y\) and \(\tilde f(\gamma)=f(\gamma^{-1})\). If \(\nu\) is faithful there is a Borel \(g>0\) on \(G\) with \(\nu(g)=1\). If \(\nu_1,\nu_2\in\mathcal E^+\) and \(\nu_1^y\le\nu_2^y\) for all \(y\), then \(\nu_1=(g\circ s)\nu_2\) for a Borel \(g\) on \(G^{(0)}\) with \(0\le g\le1\). (Definitions 1.3 and 2.1, Lemma 2.2(d),(f), Lemma 2.4 and Proposition 2.7. Proposition 2.7 gives a finite \(g\) with \(\nu_1=(g\circ s)\nu_2\); since \(\nu_1\le\nu_2\), \(g\circ s\le1\) holds \(\nu_2^y\)-almost everywhere for every \(y\), so \(\min(g,1)\) may replace \(g\).)

(R2) Transverse measures. \(\Lambda:\mathcal E^+\to[0,\infty]\) is additive, positively homogeneous, normal and of modulus \(\delta\). For \(\nu\in\mathcal E^+\), \(\Lambda_\nu(g)=\Lambda((g\circ s)\nu)\) (\(g\ge0\) finite) defines a measure \(\Lambda_\nu\) on \(G^{(0)}\), and \(\Lambda_{(g\circ s)\nu}=g\Lambda_\nu\). The modular relation: for \(\nu,\nu'\in\mathcal E^+\) and Borel \(f\ge0\) on \(G\), \[ \int\!\!\int f(\gamma^{-1})\,d\nu'^y(\gamma)\,d\Lambda_\nu(y)=\int\!\!\int f(\gamma)\,\delta(\gamma)^{-1}\,d\nu^y(\gamma) \,d\Lambda_{\nu'}(y). \tag{1.1} \] In particular the measure \(m_\nu=\Lambda_\nu\circ\nu\), \(m_\nu(f)=\int\nu^y(f)\,d\Lambda_\nu(y)\), satisfies \(\int f(\gamma^{-1})\,dm_\nu(\gamma)=\int f(\gamma)\delta(\gamma)^{-1}\,dm_\nu(\gamma)\). (Definitions 3.1 and 3.2, Lemma 3.3, and Theorem 3.5 with \(\nu\) and \(\nu'\) exchanged.)

(R3) Negligible sets. A Borel set \(A\subset G^{(0)}\) is saturated if it is a union of orbits. A saturated Borel set is \(\Lambda\)-negligible if it is \(\Lambda_\nu\)-null for every \(\nu\); if \(\nu\) is faithful this holds as soon as \(\Lambda_\nu(A)=0\). For \(\nu\in\mathcal E^+\) and a Borel set \(Z\subset G^{(0)}\) with \(\Lambda_\nu(Z)=0\), the set \([Z]_\nu=\{y:\nu^y(s^{-1}(Z))>0\}\) is a saturated Borel \(\Lambda\)-negligible set. (Definition 5.1 and Proposition 5.2.)

(R4) Random variables. A random variable of modulus \(\delta\) is a measurable functor \(F\) from \(G\) to measured spaces: a standard Borel space \(X=\bigsqcup_xF(x)\) with a Borel projection \(\pi:X\to G^{(0)}\), a Borel action \((\gamma,z)\mapsto\gamma z\) of \(G\) on \(X\), and measures \(\alpha^x\) on \(F(x)\), measurable in \(x\), such that \(X\) is an increasing union of Borel sets \(X_n\) with \(\sup_x\alpha^x(X_n)<\infty\), and \(\gamma_*\alpha^x= \delta(\gamma)\alpha^y\) for \(\gamma:x\to y\). For \(f\ge0\) on \(X\), \((\nu*f)(z)=\int f(\gamma^{-1}z)\, d\nu^{\pi(z)}(\gamma)\); it is a \(G\)-invariant function, and \(\nu*(\varphi f)=\varphi\,(\nu*f)\) for invariant \(\varphi\ge0\). \(F\) is proper if \(\nu*f=1\) for some Borel \(f\ge0\) and some (equivalently every) faithful \(\nu\); then one can take \(f>0\), and \[ \int F\,d\Lambda=\int\alpha^x(f)\,d\Lambda_\nu(x)\qquad(\nu\text{ faithful},\ \nu*f=1) \] does not depend on the choices. The functor \(L\), \(x\mapsto G^x\) with left translations, is proper; so is every sub-functor of \(\mathbb N\times L\) (a \(G\)-invariant Borel subset \(W\) of \(\mathbb N\times G\)): if \(\nu*k=1\) on \(G\), then \(f(n,\gamma)=k(\gamma)1_W(n,\gamma)\) has \(\nu*f=1_W\). (Definitions 6.1, 6.2 and 6.8, Lemma 6.3(b), Theorem 6.6(b),(c), Lemma 6.7 and Example 6.9.)

(R5) Proper homomorphisms. For a proper homomorphism \(h:G\to G'\) and a Borel homomorphism \(\delta'\) with \(\delta'\circ h=\delta\), there is an image transverse measure \(h(\Lambda)\) of modulus \(\delta'\), the pull-back \(h^*F'\) of a proper random variable \(F'\) on \(G'\) is proper, and \(\int h^*F'\,d\Lambda=\int F'\,d\,h(\Lambda)\). (Definition 7.1, Lemma 7.3 and Theorem 8.2(a),(c).)

(R6) The stable kernel of \(\delta\). Write \(c=\log\delta:G\to\mathbb R\). The groupoid \(G'=G\times\mathbb R\) has unit space \(G^{(0)}\times\mathbb R\), \(r(\gamma,s)=(r(\gamma),s)\), \(s(\gamma,s)=(s(\gamma),s+c(\gamma))\) and product \((\gamma_1,s_1)(\gamma_2,s_1+c(\gamma_1))=(\gamma_1\gamma_2,s_1)\). The map \(h(\gamma,s)=\gamma\) is a proper homomorphism, and \(G'\) has a faithful proper transverse function. Every \(\nu\in\mathcal E^+\) lifts to \(\nu'\in\mathcal E^+(G')\), \(\nu'^{(y,s)}\) the image of \(\nu^y\) under \(\gamma\mapsto(\gamma,s)\), faithful if \(\nu\) is. There is a unique transverse measure \(\Lambda'\) on \(G'\), of modulus \(\delta\circ h\), with \(\Lambda'_{\nu'}=\Lambda_\nu\times ds\) for all \(\nu\); it is \(\sigma\)-finite. (Section 9 with the homomorphism \(\psi=c\) into the unimodular group \(\mathbb R\): Lemma 9.1 and Theorem 9.2, where \(\delta_\psi=\delta\circ h\).)

1.2 Random Hilbert spaces and random operators

The following is taught in "Square-integrable representations and random operators"; the numbers in parentheses refer to that lesson.

(Q1) Coefficients. A representation \(U\) of \(G\) on a measurable field \(H=(H_x)\) gives unitaries \(U(\gamma):H_x\to H_y\) for \(\gamma:x\to y\), multiplicative and measurable. For \(\nu\in\mathcal E^+\), \(D(U,\nu)\) is the space of bounded Borel sections \(\xi\) with \[ \int|\langle\alpha,U(\gamma)\xi_{s(\gamma)}\rangle|^2\,d\nu^y(\gamma)\le c^2\|\alpha\|^2\qquad(y\in G^{(0)},\ \alpha\in H_y). \] For \(\xi\in D(U,\nu)\), \(T_\nu(\xi)_y\alpha=(\gamma\mapsto\langle\alpha,U(\gamma)\xi_{s(\gamma)}\rangle)\) is an intertwiner from \(U\) to the regular representation \(L^\nu\) on \(L^2(G^y,\nu^y)\). Its adjoint is the weak integral \(T_\nu(\xi)_y^*g=\int g(\gamma)U(\gamma)\xi_{s(\gamma)}\,d\nu^y(\gamma)\) for \(g\in L^1\cap L^2(G^y,\nu^y)\); we write \(U(g\nu)\xi\) for the section \(y\mapsto\int g(\gamma)U(\gamma)\xi_{s(\gamma)}\,d\nu^y(\gamma)\) when \(\nu(|g|)\) is bounded. For an intertwiner \(A\), \(A\xi\in D(U',\nu)\) and \(T_\nu(A\xi)=T_\nu(\xi)A^*\). We put \(\theta_\nu(\xi,\eta)=T_\nu(\xi)^*T_\nu(\eta)\); thus \[ \theta_\nu(\xi,\xi)_y=\int_{G^y}\big(U(\gamma)\xi_{s(\gamma)}\big)\big(U(\gamma)\xi_{s(\gamma)}\big)^*\,d\nu^y(\gamma) \quad\text{(weakly)},\qquad \theta_\nu(A\xi,B\eta)=A\theta_\nu(\xi,\eta)B^*. \tag{1.2} \] If \(\nu|f|\) and \(\nu'|\tilde f|\) are bounded and \(\xi\in D(U,\nu)\), then \(U(f\nu)\xi\in D(U,\nu')\). (Definitions 2.1 and 4.1, Section 2.2, and Proposition 4.2(a),(b),(c),(e),(f).)

(Q2) Square integrability. \(U\) is square integrable if, for a faithful \(\nu\), \(D(U,\nu)\) contains a countable set whose values are total in \(H_x\) for every \(x\); equivalently \(U\) is equivalent to a subrepresentation of \(\bigoplus_1^\infty L^\nu\). Subrepresentations, countable direct sums, tensor products with arbitrary representations, the representations \(L^2\circ F\) for proper random variables \(F\) (with \((U(\gamma)g)(z)= \delta(\gamma)^{1/2}g(\gamma^{-1}z)\) on \(L^2(F(x),\alpha^x)\)), and pull-backs \(h^*H\) along proper homomorphisms are square integrable. A \(\Lambda\)-random Hilbert space is a square-integrable representation up to changes on \(\Lambda\)-negligible saturated sets; its pull-back along a proper \(h\) with \(h(\Lambda_1)=\Lambda\) is a \(\Lambda_1\)-random Hilbert space. Finally, let \(\nu\) be faithful, \(S\) a countable set of bounded Borel sections and \(y\in G^{(0)}\) such that for every \(\nu^y\)-conull Borel set \(E\subset G^y\) the vectors \(U(\gamma)\xi_{s(\gamma)}\), \(\gamma\in E\), \(\xi\in S\), are total in \(H_y\). Then there is a countable family \(\mathcal D\) of bounded Borel functions \(f\) on \(G\) with \(\nu|f|\) and \(\nu|\tilde f|\) bounded, not depending on \(y\), such that the vectors \((U(f\nu)\xi)_y\), \(f\in\mathcal D\), \(\xi\in S\), are total in \(H_y\); if \(S\subset D(U,\nu)\), these sections lie in \(D(U,\nu)\) by (Q1). (Definition 4.5, Theorem 4.4, Proposition 4.6, Proposition 4.8, Corollary 4.9, Definition 6.1, Corollary 7.5(a), Proposition 2.4(b) with the family \(\mathcal D\) of Lemma 1.2(b).)

(Q3) Random operators. \(\operatorname{Hom}_\Lambda(H_1,H_2)\) is the space of bounded Borel intertwiners \((A_x)\), modulo equality off a \(\Lambda\)-negligible saturated set; \(\operatorname{End}_\Lambda(H)\) is a von Neumann algebra with separable predual. For \(\nu\in\mathcal E^+\), \(\nu(H)=\int^\oplus H_x\,d\Lambda_\nu(x)\), and \(\nu(A)\) is the decomposable operator \((A_x)\). (Definition 6.1, Theorem 7.2 and the beginning of Section 7.)

(Q4) The algebra \(W(\nu)\). For \(\nu\in\mathcal E^+\) let \(\mathcal A_\nu\) be the set of Borel functions \(f\) on \(G\) such that \(\nu|\delta^tf|\) and \(\nu|\delta^t\tilde f|\) are bounded for every \(t\in\mathbb R\), and \(f\) and \(\delta^{1/2}f\) lie in \(L^2(G,m_\nu)\). With the product \(f*_\nu g(\gamma)=\int f(\gamma')g(\gamma'^{-1}\gamma)\, d\nu^{r(\gamma)}(\gamma')\) and the involution \(f^\#=\delta^{-1}f^\flat\), \(f^\flat(\gamma)=\overline{f(\gamma^{-1})}\), it is a left Hilbert algebra in \(L^2(G,m_\nu)\), stable under \(f\mapsto\delta^{it}f\). Its left von Neumann algebra is \(W(\nu)\), generated by the operators \(\lambda(f)\) of left convolution; \(\psi_\nu\) denotes its canonical weight. Its modular conjugation is \(Jg=\delta^{-1/2}g^\flat\) and its modular operator is multiplication by \(\delta\), so \[ \sigma_t^{\psi_\nu}(\lambda(f))=\lambda(\delta^{it}f); \tag{1.3} \] indeed \(\delta^{it}(f*_\nu g)=(\delta^{it}f)*_\nu(\delta^{it}g)\) because \(\delta\) is multiplicative, so \(\delta^{it}\lambda(f)\delta^{-it}=\lambda(\delta^{it}f)\). The right convolutions \(\rho(k):g\mapsto g*_\nu k\) with \(\nu|k|\), \(\nu|\tilde k|\) bounded are bounded operators in \(W(\nu)'\). For each random Hilbert space \(H\), \(\nu(H)\) is a normal \(W(\nu)\)-module, the action \(\pi^H_\nu\) being determined by \(\pi^H_\nu(\lambda(f))\xi=U(f\nu)\xi\) for bounded \(\xi\in\nu(H)\); for \(H=L^\nu\) this is \(L^2(G,m_\nu)\) with its identity representation. There is an isometry \(V:\nu(H)\to\ell^2\otimes L^2(G,m_\nu)\) with \(V\pi^H_\nu(w)=(1\otimes w)V\) for \(w\in W(\nu)\), and every \(A\in\operatorname{Hom}_\Lambda(H_1,H_2)\) gives a \(W(\nu)\)-linear operator \(\nu(A)\), for every \(\nu\). (Section 6.2 with Proposition 6.2, Definition 6.3, Remark 6.4 and Lemma 6.5; Theorem 7.1(a),(b) and Step 3 of its proof.)

(Q5) The equivalence. If \(\nu\) is faithful, then for all random Hilbert spaces \(H_1,H_2\) the map \(A\mapsto \nu(A)\) is an isometric bijection from \(\operatorname{Hom}_\Lambda(H_1,H_2)\) onto the space of bounded \(W(\nu)\)-linear maps \(\nu(H_1)\to\nu(H_2)\). In particular \(\nu:\operatorname{End}_\Lambda(H)\to W(\nu)'\) is an isomorphism of von Neumann algebras (the commutant taken in \(\nu(H)\)). (Theorem 7.1(c) and Theorem 7.2.)

(Q6) The ideal \(\mathcal J_\nu\). The span \(\mathcal J_\nu\) of the \(\theta_\nu(\xi,\eta)\), \(\xi,\eta\in D(U,\nu)\), forms a two-sided ideal in \(\operatorname{End}_\Lambda(H)\), weakly dense if \(\nu\) is faithful. (Corollaries 4.3(a) and 7.3.)

(Q7) Points with trivial isotropy. The fibre at \(y\) of every intertwiner commutes with the isotropy group \(U(G^y_y)\). For \(\nu\) faithful there is a countable total subset \(D\subset D(U,\nu)\) such that, for every \(y\), the span of the operators \(\theta_\nu(\xi,\eta)_y\), \(\xi,\eta\in D\), is a \(*\)-algebra, weakly dense in \(\mathcal L(H_y)\) whenever \(G^y_y=\{y\}\). If \(G\) has trivial isotropy \(\Lambda\)-almost everywhere (that is, \(G^y_y=\{y\}\) off a \(\Lambda\)-negligible saturated Borel set), the centre of \(\operatorname{End}_\Lambda(H)\) consists of the classes of the families \((a(x)1_{H_x})\), \(a\) bounded, Borel and constant on orbits. (Section 5 with Proposition 5.1, and Corollary 8.1. The generation of the commutant \(U(G^y_y)'\) at points with nontrivial isotropy is only stated there, in Remark 5.2, and is not used here.)

1.3 The spatial derivative

The following is taught in "The spatial derivative"; the numbers in parentheses refer to that lesson. Fix a von Neumann algebra \(N\) with an fns weight \(\psi\) and a normal \(N\)-module \(K\), and let \(M\) be the commutant of \(N\) in \(K\).

(S1) A vector \(\zeta\in K\) is \(\psi\)-bounded if \(\|y\zeta\|\le C\|\eta_\psi(y)\|\) for \(y\in\mathfrak n_\psi\); then \(R^\psi(\zeta)\eta_\psi(y)=y\zeta\) extends to a bounded \(N\)-linear map \(K_\psi\to K\), and \(\theta^\psi(\zeta,\zeta)=R^\psi(\zeta)R^\psi(\zeta)^*\in M_+\). The \(\psi\)-bounded vectors form a dense subspace \(D(K,\psi)\). (Definition 1.1, Proposition 1.3 and Remark 1.6.)

(S2) For a normal semifinite weight \(\varphi\) on \(M\), the spatial derivative \(d\varphi/d\psi\) is positive and self-adjoint on \(K\), and \(\varphi(\theta^\psi(\zeta,\zeta))=\langle(d\varphi/d\psi)\zeta,\zeta\rangle\) for all \(\zeta\in D(K,\psi)\). Moreover \(\varphi_1\le\varphi_2\) iff \(d\varphi_1/d\psi\le d\varphi_2/d\psi\); so \(\varphi\) is determined by \(d\varphi/d\psi\). The support of \(d\varphi/d\psi\) is \(s(\varphi)\). (Definition 3.2, Propositions 3.3 and 3.4(a), Corollary 5.2.)

(S3) If \(\varphi\) is faithful, \(T=d\varphi/d\psi\) is nonsingular, \(T^{it}xT^{-it}=\sigma^\varphi_t(x)\) for \(x\in M\), and \((d\varphi_2/d\psi)^{it}=(D\varphi_2:D\varphi_1)_t(d\varphi_1/d\psi)^{it}\). (Theorem 4.3 and Theorem 5.4(a).)

(S4) Homogeneous operators. A positive self-adjoint \(T\) on \(K\) equals \(d\varphi/d\psi\) for some normal semifinite weight \(\varphi\) on \(M\) iff it is homogeneous of degree \(-1\): \[ T^{it}\sigma^\psi_t(y)=yT^{it}\qquad(y\in N,\ t\in\mathbb R). \] The weight is then unique, and faithful iff \(T\) is nonsingular. (Definition 6.1 and the remark after it, Theorem 6.2.)

2. Results used from other lessons

(B1) Operator-valued weights. Let \(N\subset P\) be von Neumann algebras, \(\varphi\) an fns weight on \(N\) and \(\rho\) an fns weight on \(P\). If \(\sigma^\rho_t(x)=\sigma^\varphi_t(x)\) for \(x\in N\), there is a unique fns operator-valued weight \(E:P_+\to\widehat N_+\) with \(\rho=\varphi\circ E\). For every fns operator-valued weight \(E\) and fns \(\varphi\), \(\sigma^{\varphi\circ E}_t\) restricts to \(\sigma^\varphi_t\) on \(N\), \(E(a^*xa)=a^*E(x)a\) (\(a\in N\)), \(E\circ\sigma^{\varphi\circ E}_t=\sigma^\varphi_t\circ E\), and \((D(\varphi_1\circ E):D(\varphi\circ E))_t=(D\varphi_1:D\varphi)_t\) for fns \(\varphi_1\). The composition of a normal weight with a normal operator-valued weight is a normal weight. These are proved in the course Modular theory and weights: existence and uniqueness in Compatible modular weights and the full positive cone, §§OE-01–OE-06; the restriction of the modular group and the cocycle identity in Modular invariants of operator-valued weights, §OM-01; normality of compositions in Finite calculus and composition of operator-valued weights, §§OVW-02 and OVW-04; \(E(a^*xa)=a^*E(x)a\) is part of the definition. For the covariance, \(E_t=\sigma^\varphi_{-t}\circ E\circ\sigma^{\varphi\circ E}_t\) is again an fns operator-valued weight, and \(\varphi\circ E_t=\varphi\circ E\) because \(\varphi\circ\sigma^\varphi_{-t}=\varphi\) and \(\varphi\circ E\) is invariant under its modular group; uniqueness gives \(E_t=E\). See also, e.g., [Haagerup 1979].

(B2) Cocycles and traces. Two fns weights with the same cocycle derivative relative to a third one are equal. An fns weight with \(\sigma^\varphi=\mathrm{id}\) is a trace. The support of a normal trace is central, and \(\varphi(x)=\varphi(exe)\) for a normal weight with support \(e\). The first statement is Reconstructing a weight from a modular cocycle, §CX-10. If \(\sigma^\varphi=\mathrm{id}\), every element is in the centralizer, so by Fixed elements and changes of density, §CZ-05 every \(a\in M\) multiplies \(\mathfrak m_\varphi\) on both sides with \(\varphi(az)=\varphi(za)\); for \(x\in\mathfrak n_\varphi\) with polar decomposition \(x=v|x|\) this gives \(\varphi(xx^*)=\varphi(v\cdot x^*xv^*)=\varphi(x^*xv^*v)=\varphi(x^*x)\), and applied to \(x^*\) it shows that the two values are finite together. The support of a normal trace is central by Traces on von Neumann algebras, Section 2 (null projections and the support), and \(\varphi(x)=\varphi(exe)\) is Finite domains, null directions, and support corners, §WS-05.

(B3) Weights given by densities. Let \(\tau\) be an fns trace on \(P\) and \(h\) a nonsingular positive self-adjoint operator affiliated with \(P\). The weight \(\tau_h(x)=\lim_{\varepsilon\to0} \tau(h_\varepsilon^{1/2}xh_\varepsilon^{1/2})\), \(h_\varepsilon=h(1+\varepsilon h)^{-1}\), is fns, and \(\sigma^{\tau_h}_t=\operatorname{Ad}h^{it}\), \((D\tau_{h_2}:D\tau_{h_1})_t=h_2^{it}h_1^{-it}\). The weight \(\tau_h\) is normal and semifinite with support \(s(h)=1\) by Fixed elements and changes of density, §CZ-08; the modular group of \(\tau\) is trivial, so \((D\tau_h:D\tau)_t=h^{it}\) by Recognizing a weight by its fixed density, §§PT-04 and PT-08, and the chain rule (Changing reference for a weight cocycle, §CH-01) gives the last formula.

(B4) Arveson spectrum [Arveson 1974]. Let \(X=E^*\) be a dual Banach space and \((\gamma_t)_{t\in\mathbb R}\) a one-parameter group of weak*-continuous linear isometries of \(X\) such that \(t\mapsto\langle\gamma_tx,a\rangle\) is continuous for all \(x\in X\), \(a\in E\). Two cases are used: a \(\sigma\)-weakly continuous action \(\alpha\) of \(\mathbb R\) on a von Neumann algebra \(M\) (\(X=M\), \(E=M_*\)); and \(\beta_t(x)=u_txv_t^*\) on \(X=B(K_2,K_1)\), for strongly continuous unitary groups \(u\) on \(K_1\) and \(v\) on \(K_2\), where \(E\) is the space of trace-class operators \(a\colon K_1\to K_2\) and \(\langle x,a\rangle=\operatorname{Tr}(xa)\). For \(f\in L^1(\mathbb R)\) let \(\gamma(f)x\in X\) be given by \(\langle\gamma(f)x,a\rangle=\int f(t)\langle\gamma_tx,a\rangle\,dt\), and put \(\hat f(p)=\int f(t)e^{ipt}dt\). The spectrum \(\operatorname{Sp}(\gamma)\) is the set of \(p\) with \(\hat f(p)=0\) for every \(f\) with \(\gamma(f)=0\); \(\operatorname{Sp}_\gamma(x)\) is defined likewise with \(\gamma(f)x=0\). Then: (i) \(\operatorname{Sp}(\gamma)\) and \(\operatorname{Sp}_\gamma(x)\) are closed, and \(\operatorname{Sp}_\gamma(x)\subset\operatorname{Sp}(\gamma)\); (ii) \(\operatorname{Sp}_\gamma(x)=\emptyset\) iff \(x=0\); (iii) if \(\hat f\) vanishes on a neighbourhood of \(\operatorname{Sp}_\gamma(x)\), then \(\gamma(f)x=0\); (iv) if the spectral measures \(e_1,e_2\) of \(u,v\) (\(u_t=\int e^{ipt}de_1(p)\), \(v_t=\int e^{ipt}de_2(p)\)) are carried by closed sets \(I_1,I_2\), then \(\operatorname{Sp}(\beta)\subset\overline{I_1-I_2}\), so \(\operatorname{Sp}_\beta(x)\subset\overline{I_1-I_2}\) for every \(x\); (v) \(\operatorname{Sp}(\gamma)\subset\{0\}\) iff \(\gamma\) is trivial; when \(X\ne0\), this says \(\operatorname{Sp}(\gamma)=\{0\}\). If an action \(\alpha\) on a von Neumann algebra satisfies \(\alpha_t(x)=u_txv_t^*\) for an element \(x\), then \(\alpha(f)x=\beta(f)x\) for every \(f\), so \(\operatorname{Sp}_\alpha(x)=\operatorname{Sp}_\beta(x)\).

Proof. We use three results on closed ideals of \(L^1(\mathbb R)\). In the lessons on harmonic analysis on locally compact abelian groups the Fourier transform is \(\int f(t)\overline{\chi(t)}\,dt\) for characters \(\chi\), and \(s\mapsto\chi_s\), \(\chi_s(t)=e^{2\pi ist}\), identifies \(\mathbb R\) with its dual group (Characters and the dual group, Theorem 3.1). Our \(\hat f(p)\) is the transform at \(\chi_{-p/2\pi}\), and \(p\mapsto\chi_{-p/2\pi}\) is a homeomorphism onto the dual group, so the results apply in our normalization. For a closed ideal \(J\) let \(h(J)\) be the set of \(p\) with \(\hat f(p)=0\) for all \(f\in J\).

• (W) If \(h(J)=\emptyset\), then \(J=L^1(\mathbb R)\) (Closed ideals of \(L^1(G)\) and Wiener's theorem, Theorem 5.3).
• (S) If \(\hat f\) vanishes on a neighbourhood of \(h(J)\), then \(f\in J\); and if \(h(J)=\{0\}\), then \(J=\{f:\int f=0\}\), because finite sets admit spectral synthesis (Tauberian theorems and spectral synthesis, Corollary 3.3).
• (U) If \(K\) is compact and \(W\supset K\) open, there is \(f\in L^1(\mathbb R)\) with \(\hat f=1\) on \(K\) and \(\operatorname{supp}\hat f\) a compact subset of \(W\) (Closed ideals of \(L^1(G)\) and Wiener's theorem, Proposition 2.1).

Since \(|\langle\gamma(f)x,a\rangle|\le\|f\|_1\|x\|\|a\|\), \(\gamma(f)x\) is well defined and \(\|\gamma(f)x\|\le\|f\|_1\|x\|\). Weak* continuity of each \(\gamma_s\) and Fubini's theorem give \(\gamma_s\gamma(f)=\gamma(f(\cdot-s))\) and \(\gamma(f)\gamma(g)=\gamma(f*g)\). Hence \(J_x=\{f:\gamma(f)x=0\}\) and \(J_\gamma=\{f:\gamma(f)=0\}\) are closed ideals, with hulls \(\operatorname{Sp}_\gamma(x)\) and \(\operatorname{Sp}(\gamma)\). If \(e_n\ge0\) has integral \(1\) and support in \((-1/n,1/n)\), then \(\gamma(e_n)x\to x\) weak*, by continuity of the scalar orbits.

(i) Each \(\hat f\) is continuous, and \(J_\gamma\subset J_x\). (ii) For \(x=0\), \(J_x=L^1(\mathbb R)\), whose hull is empty by (U). If \(\operatorname{Sp}_\gamma(x)=\emptyset\), then \(J_x=L^1(\mathbb R)\) by (W), so \(\gamma(e_n)x=0\) for all \(n\), and \(x=0\). (iii) is (S).

(v) If \(X=0\), then \(\operatorname{Sp}(\gamma)=\emptyset\) and \(\gamma\) is trivial. Let \(X\ne0\). If \(\operatorname{Sp}(\gamma)\subset\{0\}\), then \(\operatorname{Sp}(\gamma)\supset\operatorname{Sp}_\gamma(x)\ne\emptyset\) for \(x\ne0\), so \(\operatorname{Sp}(\gamma)=\{0\}\) and \(J_\gamma=\{f:\int f=0\}\) by (S). Then \(f(\cdot-s)-f\in J_\gamma\), so \(\gamma_s\gamma(f)x=\gamma(f)x\); with \(f=e_n\) and \(n\to\infty\), \(\gamma_sx=x\). Conversely, if \(\gamma\) is trivial, then \(\gamma(f)=(\int f)\,1\) and \(J_\gamma=\{f:\int f=0\}\). Its hull contains \(0\), and for \(p\ne0\), (U) with \(K=\{p\}\) and \(W=\mathbb R\setminus\{0\}\) gives \(f\in J_\gamma\) with \(\hat f(p)=1\). So \(\operatorname{Sp}(\gamma)=\{0\}\).

(iv) The pairing identifies \(B(K_2,K_1)\) with the dual of the trace class, and \(\beta_t\) is the adjoint of \(a\mapsto v_t^*au_t\), which is continuous in the trace norm, first for rank-one \(a\) and then by density; so \(\beta\) is of the kind considered. Let \(F=\overline{I_1-I_2}\) and suppose that \(\hat f=0\) on \(F\). Put \(P_n=e_1([-n,n])\) and \(Q_n=e_2([-n,n])\). Divide \([-n,n]\) into finitely many Borel intervals of length at most \(1/m\), let \(E_i\) and \(F_j\) be the nonzero spectral projections of \(e_1\) and \(e_2\) of these intervals, and choose \(a_i\in I_1\) and \(b_j\in I_2\) in the corresponding intervals, which is possible because \(e_1,e_2\) vanish off \(I_1,I_2\). For \(U(t)=\sum_ie^{ita_i}E_i\) and \(V(t)=\sum_je^{itb_j}F_j\), the spectral theorem gives \(\|U(t)-u_tP_n\|\le\min(2,|t|/m)\) and \(\|V(t)-v_tQ_n\|\le\min(2,|t|/m)\), and \[ \int f(t)U(t)xV(t)^*\,dt=\sum_{i,j}\hat f(a_i-b_j)\,E_ixF_j=0, \] because \(a_i-b_j\in F\). Since \(P_n\) commutes with \(u_t\) and \(Q_n\) with \(v_t\), \(\int f(t)u_tP_nxQ_nv_t^*\,dt=P_n\beta(f)xQ_n\), and dominated convergence as \(m\to\infty\) gives \(P_n\beta(f)xQ_n=0\). Letting \(n\to\infty\), \(\beta(f)x=0\) for every \(x\), that is, \(\beta(f)=0\). If \(p\notin F\), (U) with \(K=\{p\}\) and \(W=\mathbb R\setminus F\) gives such an \(f\) with \(\hat f(p)=1\), so \(p\notin\operatorname{Sp}(\beta)\). \(\square\)

(B5) Direct integrals. Let \(\mu\) be a \(\sigma\)-finite measure on a standard Borel space \(Y\) and \(\mathcal D\) the diagonal algebra of \(\int^\oplus K_y\,d\mu\). Its commutant is the algebra of decomposable operators. A positive self-adjoint operator whose unitary group and support commute with \(\mathcal D\) is a direct integral \(\int^\oplus T_y\,d\mu\) of a measurable field of positive self-adjoint operators. If a von Neumann algebra \(R\supset\mathcal D\) is generated by \(\mathcal D\) and countably many decomposable operators \(b^n\), then \(R=\int^\oplus R_y\,d\mu\) with \(R_y\) generated by the \(b^n_y\), and \(R'=\int^\oplus R_y'\,d\mu\), \(R\cap R'= \int^\oplus(R_y\cap R_y')\,d\mu\). The commutant of the diagonal algebra is Decomposable operators and the diagonal algebra, Theorem 5.1; the algebra generated by \(\mathcal D\) and the \(b^n\) is \(\int^\oplus R_y\,d\mu\) by Lemma 5.1 of Direct integrals of von Neumann algebras, with commutant and centre given by Theorems 3.2(1) and 4.3 there. The statement on positive self-adjoint operators is Unbounded decomposable operators and measurable spectral decomposition, Corollary 4.2, for an arbitrary \(\sigma\)-finite base; there a field of self-adjoint operators is measurable when its Cayley transforms are (Definition 3.1), and every bounded Borel function of \(\int^\oplus T_y\,d\mu\) is the direct integral of the same function of the \(T_y\) (Theorem 3.6), in particular \(\big(\int^\oplus T_y\,d\mu\big)^{it}=\int^\oplus T_y^{it}\,d\mu\).

(B6) Measurable spectral decomposition. Let \(\mu\) be a \(\sigma\)-finite measure on a standard Borel space \(Y\) and \((A_y)\) a measurable field of self-adjoint operators on a measurable field \((K_y)\) whose spectral measures are absolutely continuous. There are a measurable field \((K_{(y,s)})\) over \(Y\times\mathbb R\) and a measurable field of unitaries \(K_y\to\int^\oplus K_{(y,s)}\,ds\) (defined for \(\mu\)-almost every \(y\)) carrying \(A_y\) to multiplication by \(s\). If \((\tilde A_y)\) on \((\tilde K_y)\) is a second such field, decomposed in the same way, a measurable field \((B_y)\) of bounded operators \(B_y:K_y\to\tilde K_y\) with \(B_yg(A_y)=g(\tilde A_y)B_y\) for all bounded Borel \(g\) is carried to \((\int^\oplus B'_{(y,s)}\,ds)_y\) for a measurable field of operators \(B'_{(y,s)}:K_{(y,s)}\to\tilde K_{(y,s)}\) over \(Y\times\mathbb R\). This is proved, with Lebesgue measure as the dominating measure \(\kappa\) and for an arbitrary \(\sigma\)-finite base, in Unbounded decomposable operators and measurable spectral decomposition, Theorems 8.1 and 8.3; the measurable fields over \(Y\times\mathbb R\) and the spaces \(\int^\oplus K_{(y,s)}\,ds\) are those of Theorem 7.1 there, and the second statement holds for any two decompositions of this kind, with \(\|B'_{(y,s)}\|\le\|B_y\|\).

(B7) Almost homomorphisms [Ramsay 1971]. Let \(\mathcal G\) be a groupoid whose Borel structure is standard, \(\kappa\) a faithful proper transverse function on \(\mathcal G\) and \(\mu\) a \(\sigma\)-finite measure on \(\mathcal G^{(0)}\) such that \(m=\mu\circ\kappa\) and its image under \(\gamma\mapsto\gamma^{-1}\) have the same null sets. Let \(\pi\) be a Borel map from \(\mathcal G\) to a Polish group such that \(\pi(\gamma_1\gamma_2)=\pi(\gamma_1) \pi(\gamma_2)\) for almost every composable pair, for the measure \(\int\!\!\int\varepsilon_{(\gamma_1,\gamma_2)}\, d\kappa^{s(\gamma_1)}(\gamma_2)\,dm(\gamma_1)\) on \(\mathcal G^{(2)}\). Then there are a \(\mu\)-conull saturated Borel set \(Y\subset\mathcal G^{(0)}\) and a Borel homomorphism \(\pi'\) from the reduction \(\mathcal G_Y=r^{-1}(Y)\) to the Polish group with \(\pi'=\pi\) \(m\)-almost everywhere on \(\mathcal G_Y\). This is Almost homomorphisms of measured groupoids, Theorem 5.1. Ramsay's theory of virtual groups works with reductions to conull Borel sets that need not be saturated, as Connes points out [Connes 1979, Introduction]; that lesson reaches a saturated set by joining each unit to the conull set with a Borel choice of arrows.

(B8) Projections. In a von Neumann algebra: the left and right supports \(l(x)\), \(r(x)\) of an element are equivalent; \(p\vee q-q\sim p-p\wedge q\) (parallelogram law); two projections with \(\|p-q\|<1\) are equivalent, and so are \(1-p\) and \(1-q\). For closed subspaces with projections \(P_1,P_2\), \(\|P_1-P_2\|= \max(\|(1-P_2)P_1\|,\|(1-P_1)P_2\|)\), and \(\|(1-P_2)P_1\|\) is the supremum over unit vectors of \(P_1\) of the distance to the range of \(P_2\). If \(b\in M\) is injective on the range of a projection \(q\in M\), then \(q\) is equivalent to the projection onto the closure of \(bqH\) (polar decomposition of \(bq\)); if \(b\) is bounded below on \(qH\), this range is closed. The inverse of an element of \(fMe\) mapping \(eH\) bijectively onto \(fH\) lies in \(eMf\). The supports are equivalent through the partial isometry of the polar decomposition, which lies in \(M\) by The double commutant theorem, Proposition 7.2; this also gives the statement on \(bq\). The parallelogram law is Projections and types of von Neumann algebras, Proposition 4.4. If \(\|p-q\|<1\), then \(x=qp+(1-q)(1-p)\) satisfies \(x^*x=1-(p-q)^2\), so it is invertible, and \(|x|\) commutes with \(p\); its unitary polar part \(u\in M\) satisfies \(upu^*=xpx^{-1}=q\), which gives both equivalences. An element of \(fMe\) mapping \(eH\) bijectively onto \(fH\) has a bounded inverse there (Hahn–Banach, Baire and the basic theorems on Banach spaces, the open mapping theorem); extended by \(0\) on \((1-f)H\), the inverse commutes with \(M'\), so it lies in \(M\), and in \(eMf\). The two norm formulas are elementary Hilbert-space geometry.

(B9) Groups. Let \(\Gamma\) be a unimodular locally compact group with Haar measure \(dg\), also written \(m\). No countability assumption is made in this item.

(a) Every nonzero \(\sigma\)-finite left-invariant measure on the Borel sets of \(\Gamma\) is \(c\,dg\) for a unique \(c>0\).

(b) The right von Neumann algebra \(R(\Gamma)=\lambda(\Gamma)'\) carries the Plancherel trace \(\tau'\): call \(\eta\in L^2(\Gamma)\) right bounded if \(a\mapsto a*\eta\) (\(a\in C_c(\Gamma)\)) extends to a bounded operator \(\rho(\eta)\) on \(L^2(\Gamma)\); then \(\tau'(\rho(\eta)^*\rho(\eta))=\|\eta\|^2\), and \(\tau'(x^*x)=\infty\) for every other \(x\in R(\Gamma)\). It is an fns trace.

(c) Let \(\pi\) be an irreducible strongly continuous unitary representation of \(\Gamma\) on \(K_\pi\ne0\), with coefficients \(c_{\xi,\eta}(g)=\langle\pi(g)\xi,\eta\rangle\). The following are equivalent: some \(c_{\xi_0,\eta_0}\) with \(\xi_0,\eta_0\ne0\) is square integrable; every \(c_{\xi,\eta}\) is square integrable; \(\pi\) is equivalent to a subrepresentation of the left regular representation \(\lambda\). Such a \(\pi\) is called square integrable, and there is a unique \(d_\pi>0\), the formal degree, with \[ \int c_{\xi,\eta}(g)\overline{c_{\xi',\eta'}(g)}\,dg=d_\pi^{-1}\langle\xi,\xi'\rangle\overline{\langle\eta,\eta'\rangle}, \qquad\text{in particular}\qquad\int|c_{\xi,\eta}|^2\,dg=d_\pi^{-1}\|\xi\|^2\|\eta\|^2. \]

For second countable \(\Gamma\), (a) is Regular representations, Fourier algebra and Fourier–Stieltjes coefficients, §PF-39, and Section 10 uses (a) only in that case. The proofs below, of (a) in general and of (b) and (c), use these results of Haar measure on locally compact groups: the Riesz representation theorem and inner regularity on sets of finite measure (Theorem 2.2 and Proposition 2.3(1)); image measures and density of \(C_c\) in \(L^p\) (Proposition 3.1); an open \(\sigma\)-compact subgroup (Proposition 7.2); positivity of Haar measure on nonempty open sets and its uniqueness (Proposition 9.1 and Theorem 9.2); invariance of \(dg\) under right translations and under \(g\mapsto g^{-1}\), \(\Gamma\) being unimodular (Theorem 10.1 and Theorem 11.1); the bound \(\|\xi*k\|_2\le\|k\|_1\|\xi\|_2\) and continuity of translation in \(L^1\) (Theorem 14.2(4),(6)); and approximate identities (Theorem 15.1).

Proof of (a). Let \(\nu\) be such a measure. We use two facts. A \(\sigma\)-finite measure has at most countably many pairwise disjoint measurable sets of positive measure: if sets \(E_n\) of finite measure cover the space, each such set meets some \(E_n\) in measure at least \(1/k\), and for fixed \(n,k\) there are only finitely many. Next, by a theorem of Ulam, there is no countably additive probability measure \(P\) on all subsets of the first uncountable ordinal \(\omega_1\) with \(P(\{\beta\})=0\) for every \(\beta\). Indeed, choose for each countable ordinal \(\beta\ge1\) a surjection \(e_\beta\colon\mathbb N\to\beta\), and put \(A_{\alpha,n}=\{\beta>\alpha:e_\beta(n)=\alpha\}\). For fixed \(n\) these sets are pairwise disjoint, so \(P(A_{\alpha,n})>0\) for only countably many \(\alpha\); choose \(\alpha\) outside all these countable sets. Then \(\bigcup_nA_{\alpha,n}=\{\beta:\beta>\alpha\}\) has probability \(0\), and so does its countable complement, a contradiction.

Step 1: \(\Gamma\) is \(\sigma\)-compact. This step, which adapts the argument of [Erdős–Mauldin 1976], does not use unimodularity. Suppose \(\Gamma\) is not \(\sigma\)-compact. Starting from an open \(\sigma\)-compact subgroup \(H_0\), build by transfinite recursion open \(\sigma\)-compact subgroups \(H_\alpha\), \(\alpha<\omega_1\), with \(H_\alpha\subsetneq H_{\alpha+1}\) and \(H_\lambda=\bigcup_{\alpha<\lambda}H_\alpha\) for limit \(\lambda\). For \(H_{\alpha+1}\) take the subgroup generated by \(H_\alpha\) and some \(g_\alpha\notin H_\alpha\), which exists because \(H_\alpha\ne\Gamma\); it is the union of the \(\sigma\)-compact sets \(W^n\), \(W=H_\alpha\cup\{g_\alpha,g_\alpha^{-1}\}\), and it is open. An increasing countable union of open \(\sigma\)-compact subgroups is again one. Put \(H=\bigcup_\alpha H_\alpha\). For each \(\alpha\), \(H_\alpha\) has uncountably many left cosets in \(H\): countably many coset representatives would all lie in one \(H_\beta\), since countably many countable ordinals have a countable upper bound, and then \(H=H_\beta\ne H_{\beta+1}\). Let \(R\) be a set of representatives of the right cosets of \(H\) in \(\Gamma\), and \(S_\alpha=H_\alpha R\). It is a union of right cosets of the open subgroup \(H_\alpha\), and so is its complement; so \(S_\alpha\) is clopen. If \(M_\alpha\subset H\) represents the left cosets of \(H_\alpha\) in \(H\), the sets \(xS_\alpha\), \(x\in M_\alpha\), are pairwise disjoint: \(xhr=x'h'r'\) with \(h,h'\in H_\alpha\) gives \(Hr=Hr'\), so \(r=r'\), and then \(xH_\alpha=x'H_\alpha\). They all have measure \(\nu(S_\alpha)\), and there are uncountably many, so \(\nu(S_\alpha)=0\). The sets \(S_\alpha\) increase, \(S_\lambda=\bigcup_{\alpha<\lambda}S_\alpha\) for limit \(\lambda\), and \(\bigcup_\alpha S_\alpha=HR=\Gamma\). So \(T_0=S_0\) and \(T_{\alpha+1}=S_{\alpha+1}\setminus S_\alpha\) (\(T_\lambda=\emptyset\) for limit \(\lambda\)) partition \(\Gamma\) into clopen \(\nu\)-null sets indexed by \(\omega_1\), and every union of them is open, hence Borel. For a Borel set \(E\) with \(0<\nu(E)<\infty\), \(P(A)=\nu\bigl(E\cap\bigcup_{i\in A}T_i\bigr)/\nu(E)\) contradicts Ulam's theorem.

Step 2: compact zero sets. A zero set is a set \(q^{-1}(0)\) with \(q\colon\Gamma\to[0,\infty)\) continuous. For every Borel \(E\) with \(m(E)<\infty\), \(m(E)=\sup\{m(C):C\subset E\text{ a compact zero set}\}\); this is a theorem of Halmos, in the form of [Fremlin, 443M]. Given \(r<m(E)\), choose a compact \(K\subset E\) with \(m(K)>r\), and let \(C\) be the set of points of \(K\) all of whose neighbourhoods \(O\) satisfy \(m(K\cap O)>0\); it is compact. Every compact subset of \(K\setminus C\) is covered by finitely many open sets \(O\) with \(m(K\cap O)=0\), so \(m(K\setminus C)=0\) by inner regularity, and \(m(C)=m(K)>r\). The function \(\varphi(s)=m(C\,\triangle\,sC)=\|1_C-1_{sC}\|_1\) is continuous; choose open neighbourhoods \(U_n\) of the identity with \(\varphi<2^{-n}\) on \(U_n\). Then \(C=\bigcap_nU_nC\). Indeed, let \(x=u_nz_n\) with \(u_n\in U_n\), \(z_n\in C\), and suppose \(x\notin C\). A subnet of \((z_n)\) converges to some \(z\in C\); along it \(u_n\to u=xz^{-1}\), and \(\varphi(u)=0\), so \(m(C\setminus u^{-1}C)=0\) by left invariance. But \(z\in C\setminus u^{-1}C\), and the open set \(\Gamma\setminus u^{-1}C\) contains \(z\), so \(m(C\setminus u^{-1}C)=m(K\setminus u^{-1}C)>0\), a contradiction. Choose \(\chi_n\in C_c(\Gamma)\) with \(0\le\chi_n\le1\), \(\chi_n=1\) on \(C\) and support in the open set \(U_nC\). Then \(q=\sum_n2^{-n}(1-\chi_n)\) is continuous with zero set exactly \(C\).

Step 3: \(\nu\) is finite on compact sets. By Step 1, \(\Gamma\) is the union of an increasing sequence of compact sets \(K_n\). Every continuous function on \(\Gamma\times\Gamma\) is then measurable for the product of the Borel \(\sigma\)-algebras, since on each \(K_n\times K_n\) it is a uniform limit of finite sums of products \(a(s)b(t)\) of continuous functions (Stone–Weierstrass). Since \(\nu\) is \(\sigma\)-finite, some Borel set \(E\) with \(\nu(E)<\infty\) and some \(K_n\) satisfy \(m(K_n\cap E)>0\); Step 2 gives a compact zero set \(C=q^{-1}(0)\subset K_n\cap E\) with \(m(C)>0\) and \(\nu(C)<\infty\). Let \(b\in C_c(\Gamma)\) be nonnegative and nonzero, and \(h(x)=\int b(t)1_C(tx)\,dm(t)\). The integrand is product measurable, since \((t,x)\mapsto q(tx)\) is continuous, and Tonelli's theorem for the \(\sigma\)-finite measures \(m\) and \(\nu\), with left invariance, gives \[ \int h\,dm=m(C)\int b\,dm>0,\qquad\int h\,d\nu=\nu(C)\int b\,dm<\infty. \] By right invariance of \(m\), \(h(x)=\int_Cb(yx^{-1})\,dm(y)\); it is continuous, because \(b(yx^{-1})\to b(yx_0^{-1})\) uniformly in \(y\in C\) as \(x\to x_0\). So \(h\ge\varepsilon\) on a nonempty open set \(U\) for some \(\varepsilon>0\), and \(\nu(U)\le\varepsilon^{-1}\int h\,d\nu<\infty\). Every compact set is covered by finitely many left translates of \(U\), so it has finite \(\nu\)-measure.

Step 4: \(\nu=c\,m\). By Step 3, \(f\mapsto\int f\,d\nu\) is a positive linear functional on \(C_c(\Gamma)\); let \(\mu\) be the Radon measure representing it. For \(s\in\Gamma\), \(E\mapsto\mu(sE)\) is a Radon measure representing the same functional, by left invariance of \(\nu\), so \(\mu\) is left invariant by the uniqueness in the Riesz theorem. It is nonzero: some \(K_n\) has \(\nu(K_n)>0\), and a function of \(C_c(\Gamma)\) that is \(\ge1_{K_n}\) has positive \(\nu\)-integral. By uniqueness of Haar measure, \(\mu=c\,m\) with \(c>0\). For a compact zero set \(C=q^{-1}(0)\) and \(\chi\in C_c(\Gamma)\) with \(0\le\chi\le1\) and \(\chi=1\) on \(C\), the functions \(\chi\max(0,1-nq)\) decrease to \(1_C\), and dominated convergence gives \(\nu(C)=c\,m(C)\). For a relatively compact Borel set \(A\), Step 2 gives \(\nu(A)\ge c\,m(A)\). Choose \(\psi\in C_c(\Gamma)\) with \(0\le\psi\le1\) and \(\psi=1\) on \(\bar A\); then \(Z=\{\psi\ge1/2\}\) is a compact zero set containing \(A\), and \(\nu(A)=\nu(Z)-\nu(Z\setminus A)\le c\,m(Z)-c\,m(Z\setminus A)=c\,m(A)\). So \(\nu=c\,m\) on relatively compact Borel sets, and on all Borel sets by intersecting with the \(K_n\). Since \(m\ne0\), \(c\) is unique. \(\square\)

Proof of (b). By Haar convolution from both sides, §§HG-01–HG-03, \(C_c(\Gamma)\) with convolution and the involution \(a^*(g)=\overline{a(g^{-1})}\) is a left Hilbert algebra in \(L^2(\Gamma)\). Its left von Neumann algebra is \(\lambda(\Gamma)''\): the operator \(a*\cdot\) is the weak integral \(\int a(g)\lambda(g)\,dg\), so it lies in \(\lambda(\Gamma)''\), and for an approximate identity \((a_U)\), \((a_U(s^{-1}\cdot))*\cdot=\lambda(s)(a_U*\cdot)\to\lambda(s)\) strongly. So its commutant is \(R(\Gamma)\), and Weights and the Hilbert spaces of multiplication, §WH-12 gives an fns weight \(\tau'\) on \(R(\Gamma)\) whose finite ideal consists of the \(\rho(\eta)\), \(\eta\) right bounded for the original algebra \(C_c(\Gamma)\), with \(\tau'(\rho(\eta)^*\rho(\eta))=\|\eta\|^2\). It is a trace. Let \(J\) be the antiunitary \((J\eta)(g)=\overline{\eta(g^{-1})}\). For \(a,b\in C_c(\Gamma)\) and \(\eta\in L^2(\Gamma)\), Fubini's theorem gives \(\langle a*\eta,b\rangle=\langle a,b*J\eta\rangle\). If \(\eta\) is right bounded, then \(|\langle a,b*J\eta\rangle|\le\|\rho(\eta)\|\|a\|\|b\|\), so \(J\eta\) is right bounded and \(\rho(\eta)^*=\rho(J\eta)\). Hence the finite ideal is closed under adjoints, and for \(x=\rho(\eta)\), \(\tau'(xx^*)=\|J\eta\|^2=\|\eta\|^2=\tau'(x^*x)\); for every other \(x\) both values are infinite. For a unitary \(u\in R(\Gamma)\) and \(b\in R(\Gamma)_+\), \(x=ub^{1/2}\) gives \(\tau'(ubu^*)=\tau'(b)\). \(\square\)

Proof of (c). Since \(\Gamma\) is unimodular, \(dg\) is invariant under \(g\mapsto gs\) and \(g\mapsto g^{-1}\), so these maps preserve square integrability. An operator commuting with \(\pi(\Gamma)\) is scalar (Schur's lemma): the spectral projections of its real and imaginary parts have invariant ranges. Fix \(\eta\ne0\), let \(D_\eta\) be the set of \(\xi\in K_\pi\) for which \(T_\eta\xi\colon g\mapsto\langle\xi,\pi(g)\eta\rangle=c_{\xi,\eta}(g^{-1})\) is square integrable, and regard \(T_\eta\) as an operator from \(D_\eta\) to \(L^2(\Gamma)\).

(1) If \(D_\eta\ne0\), then \(D_\eta=K_\pi\) and \(\|T_\eta\xi\|_2^2=c_\eta\|\xi\|^2\) with \(c_\eta>0\). From \(T_\eta\pi(s)\xi=\lambda(s)T_\eta\xi\), \(D_\eta\) is invariant under \(\pi\), hence dense by irreducibility. \(T_\eta\) is closed: if \(\xi_n\to\xi\) and \(T_\eta\xi_n\to F\) in \(L^2\), the coefficients converge uniformly to \(T_\eta\xi\), and a subsequence with \(\sum_k\|T_\eta\xi_{n_k}-F\|_2^2<\infty\) converges to \(F\) almost everywhere, so \(F=T_\eta\xi\). By Normal products and closed operator graphs, §3, \(A=T_\eta^*T_\eta\) is a positive self-adjoint operator with \(D(A^{1/2})=D_\eta\) and \(\|A^{1/2}\xi\|=\|T_\eta\xi\|\). The relation \(T_\eta\pi(s)=\lambda(s)T_\eta\), with equal domains, gives \(\pi(s)^*A\pi(s)=A\), so \((1+A)^{-1}\) commutes with \(\pi\) and is a scalar \(a\in(0,1]\). Hence \(A=(a^{-1}-1)1\) is bounded, \(D_\eta=K_\pi\), and \(c_\eta=a^{-1}-1\). Finally \(c_\eta>0\): \(T_\eta\eta\) is continuous with value \(\|\eta\|^2\) at the identity, and a nonzero continuous function does not vanish almost everywhere, Haar measure being positive on nonempty open sets.

(2) Then \(c_\eta^{-1/2}T_\eta\) is an isometry \(K_\pi\to L^2(\Gamma)\) intertwining \(\pi\) and \(\lambda\), so \(\pi\) is equivalent to a subrepresentation of \(\lambda\).

(3) One square-integrable coefficient makes all of them square integrable. If \(c_{\xi_0,\eta_0}\) is square integrable, then \(\xi_0\in D_{\eta_0}\), so \(T_{\eta_0}\) is everywhere defined by (1). For \(\xi\ne0\), \(T_\xi\eta_0(g)=\overline{T_{\eta_0}\xi(g^{-1})}\) is square integrable, so \(\eta_0\in D_\xi\) and \(D_\xi=K_\pi\). So every coefficient is square integrable, and \(\pi\) is a subrepresentation of \(\lambda\) by (2).

(4) Subrepresentations of \(\lambda\). Let \(\pi\) be the restriction of \(\lambda\) to a closed invariant subspace \(H\subset L^2(\Gamma)\) with projection \(P\), which commutes with \(\lambda\). By density of \(C_c(\Gamma)\) choose \(h\in C_c(\Gamma)\) with \(\eta=Ph\ne0\). For \(\xi\in H\), \(\langle\xi,\lambda(g)\eta\rangle=\langle\xi,\lambda(g)h\rangle=(\xi*h^*)(g)\) with \(h^*(s)=\overline{h(s^{-1})}\), and \(\|\xi*h^*\|_2\le\|h\|_1\|\xi\|_2\). So \(D_\eta=H\), and the coefficient \(c_{\eta,\eta}\), which is nonzero, is square integrable; by (3) all coefficients are.

(5) The formal degree. For \(\xi,\eta\ne0\), \(\|T_\eta\xi\|_2=\|T_\xi\eta\|_2\), by the identity in (3) and invariance under inversion; so \(c_\eta\|\xi\|^2=c_\xi\|\eta\|^2\). With a unit vector \(e\) and \(c=c_e>0\), this gives \(c_\eta=c\|\eta\|^2\), that is, \(\int|c_{\xi,\eta}|^2\,dg=c\|\xi\|^2\|\eta\|^2\). Put \(d_\pi=c^{-1}\). Polarization in \(\xi\) for fixed \(\eta\), and then in \(\eta\), gives the formula for four vectors, and evaluation at unit vectors gives uniqueness of \(d_\pi\). \(\square\)

Square-integrable representations and the formal degree are treated for every locally compact group in [Duflo–Moore 1976].

(B10) Left Hilbert algebras. Let \(\mathcal A\) be a left Hilbert algebra in a Hilbert space \(\mathfrak h\), with left von Neumann algebra \(L=\lambda(\mathcal A)''\) and canonical weight \(\psi\), an fns weight on \(L\). (i) The GNS space of \(\psi\) is \(\mathfrak h\), with \(\eta_\psi(\lambda(a))=a\) for \(a\in\mathcal A\). (ii) Call \(\zeta\in\mathfrak h\) right bounded if \(a\mapsto\lambda(a)\zeta\) (\(a\in\mathcal A\)) extends to a bounded operator \(\pi_r(\zeta)\) on \(\mathfrak h\). Then \(y\zeta=\pi_r(\zeta)\eta_\psi(y)\) for every \(y\in\mathfrak n_\psi\).

3. Random operators of degree \(\alpha\)

A closed densely defined operator \(T\) is determined by the pair of bounded operators \((u,(1+T^*T)^{-1})\), where \(T=u|T|\) is the polar decomposition. A family \((T_x)_{x\in G^{(0)}}\) of closed densely defined operators \(T_x:H_x\to H'_x\) between measurable fields is measurable if both bounded families \((u_x)\) and \(((1+T_x^*T_x)^{-1})\) are measurable. For positive self-adjoint \(T_x\) this is the same as measurability of \(x\mapsto T_x^{it}\) for every \(t\), because \(T^{it}\) and \((1+T)^{-1}\) are obtained from each other by Borel functional calculus, and a Borel function of a measurable field is measurable.

Definition 3.1. Let \((H,U)\) and \((H',U')\) be square-integrable representations and \(\alpha\in\mathbb R\). A measurable family \(T=(T_x)\) of closed densely defined operators \(T_x:H_x\to H'_x\) has degree \(\alpha\) if for every \(\gamma:x\to y\) \[ U'(\gamma)T_x=\delta(\gamma)^{\alpha}\,T_y\,U(\gamma) \tag{3.1} \] as closed operators; in particular \(U(\gamma)\operatorname{dom}T_x=\operatorname{dom}T_y\). A random operator of degree \(\alpha\) is a class of such families modulo equality off a \(\Lambda\)-negligible saturated Borel set (the relation (3.1) is only required off such a set). It is positive if \(H=H'\) and every \(T_x\) is positive self-adjoint.

Random operators of degree \(0\) that are bounded are exactly the elements of \(\operatorname{Hom}_\Lambda(H,H')\). For each \(t\in\mathbb R\) let \(H^{(t)}\) denote the field \(H\) with the representation \(U^{(t)}(\gamma)= \delta(\gamma)^{it}U(\gamma)\). It is square integrable, since \(|\langle\alpha,U^{(t)}(\gamma)\xi_x\rangle|= |\langle\alpha,U(\gamma)\xi_x\rangle|\) gives \(D(U^{(t)},\nu)=D(U,\nu)\).

Lemma 3.2. Let \(T=(T_x)\) be a measurable family of positive self-adjoint operators on \(H\). The following are equivalent.

(a) \(T\) has degree \(1\).

(b) For every \(t\in\mathbb R\) and \(\gamma:x\to y\), \(U(\gamma)T_x^{it}U(\gamma)^*=\delta(\gamma)^{it}T_y^{it}\).

(c) For every \(t\), the family \((T_x^{it})\) is an intertwiner from \(U^{(t)}\) to \(U\): \(T_y^{it}\,\delta(\gamma)^{it}U(\gamma)=U(\gamma)T_x^{it}\).

In this case the support family \(s(T)=(s(T_x))\) is a projection of degree \(0\), that is, an element of \(\operatorname{End}_G(H)\).

Proof. (a) says \(U(\gamma)T_xU(\gamma)^*=\delta(\gamma)T_y\). Conjugation by a unitary commutes with the functional calculus, and \((cS)^{it}=c^{it}S^{it}\) for \(c>0\) (with \(0^{it}=0\) on the kernel), so (a) implies (b). Conversely, two positive self-adjoint operators \(S_1,S_2\) with \(S_1^{it}=S_2^{it}\) for all \(t\) have the same support (\(t=0\)) and the same unitary group on it, hence are equal by the uniqueness in Stone's theorem; applied to \(S_1=U(\gamma)T_xU(\gamma)^*\), \(S_2=\delta(\gamma)T_y\), (b) implies (a). (b) and (c) are the same equation multiplied on the right by \(U(\gamma)\). At \(t=0\), (b) reads \(U(\gamma)s(T_x)U(\gamma)^*=s(T_y)\). \(\square\)

Examples 3.3. (a) If \(\delta=1\), the identity family \(1=(1_{H_x})\) is a positive random operator of degree \(1\). This is the case of Section 10.

(b) Let \(F\) be a proper random variable and \(H=L^2\circ F\) (see (Q2)). Let \(\rho>0\) be a Borel function on \(X=\bigsqcup F(x)\) with \(\rho(\gamma^{-1}z)=\delta(\gamma)\rho(z)\) for \(z\in F(y)\), \(\gamma:x\to y\), and let \(M(\rho)_x\) be multiplication by \(\rho\) on \(L^2(F(x),\alpha^x)\). Since \(U(\gamma)M(g)_xU(\gamma)^*\) is multiplication by \(z\mapsto g(\gamma^{-1}z)\) on \(F(y)\) for every Borel \(g\), \[ U(\gamma)M(\rho)_xU(\gamma)^*=M(\rho\circ\gamma^{-1})_y=\delta(\gamma)M(\rho)_y, \] so \(M(\rho)\) is a positive random operator of degree \(1\). We call such operators diagonal.

(c) In particular, for \(\nu\in\mathcal E^+\) the regular representation \(L^\nu\) is \(L^2\circ F\) for the random variable \(x\mapsto(G^x,\delta^{-1}\nu^x)\): indeed \(\gamma_*(\delta^{-1}\nu^x)=\delta(\gamma)\delta^{-1}\nu^y\) by left invariance and multiplicativity, and \(g\mapsto\delta^{1/2}g\) is a unitary from \(L^2(G^x,\nu^x)\) onto \(L^2(G^x,\delta^{-1}\nu^x)\) that carries \(L^\nu(\gamma)\) to \(g\mapsto\delta(\gamma)^{1/2}g(\gamma^{-1}\,\cdot)\). On \(L^\nu\) itself, multiplication by \(\delta^{-1}\) has degree \(1\): \(\delta(\gamma^{-1}\gamma')^{-1}= \delta(\gamma)\delta(\gamma')^{-1}\).

Lemma 3.4 (rigidity). Let \(\nu\in\mathcal E^+\) be faithful and let \(T_1,T_2\) be families of degree \(\alpha\) between \(H\) and \(H'\), each satisfying (3.1) on a saturated Borel set \(Y\) with \(\Lambda\)-negligible complement. If \(T_{1,x}=T_{2,x}\) for \(\Lambda_\nu\)-almost every \(x\), then \(T_1=T_2\) as random operators.

Proof. Let \(Z\subset Y\) be the set where \(T_{1,x}\neq T_{2,x}\). Using the bounded pairs \((u,(1+T^*T)^{-1})\) and a countable family of Borel sections whose values are dense in every fibre, \(Z\) is Borel. By (3.1), for \(\gamma:x\to y\) in \(Y\) we have \(T_{j,y}=\delta(\gamma)^{-\alpha}U'(\gamma)T_{j,x}U(\gamma)^*\), so \(x\in Z\) iff \(y\in Z\): \(Z\) is saturated. Since \(\Lambda_\nu(Z)=0\) and \(\nu\) is faithful, \(Z\) is \(\Lambda\)-negligible by (R3). \(\square\)

The next lemma turns an operator on the direct integral \(\nu(H)\) into a random operator of degree \(1\). Recall from (Q4) that \(\nu(H^{(t)})\) is the Hilbert space \(\nu(H)\) on which \(\lambda(f)\) acts by \(U^{(t)}(f\nu)= U(\delta^{it}f\nu)\).

Lemma 3.5. Let \(\nu\in\mathcal E^+\) be faithful, and let \(S\) be positive and self-adjoint on \(\nu(H)\), such that, for every \(t\), \(S^{it}\) is a \(W(\nu)\)-linear map from \(\nu(H^{(t)})\) to \(\nu(H)\). Then there is a unique positive random operator \(T\) of degree \(1\) with \(\nu(T)=S\), where \(\nu(T)=\int^\oplus T_x\,d\Lambda_\nu(x)\).

Proof. Uniqueness follows from Lemma 3.4. For each rational \(t\), (Q5) applied to \(H^{(t)}\) and \(H\) gives \(W^{(t)}\in\operatorname{Hom}_\Lambda(H^{(t)},H)\) with \(\nu(W^{(t)})=S^{it}\). Choose Borel representatives which are genuine intertwiners off a \(\Lambda\)-negligible saturated set; discarding countably many such sets we may assume they are intertwiners everywhere.

Twisting both sides by \(\delta^{is}\), \(W^{(t)}\) is also an intertwiner from \(H^{(s+t)}\) to \(H^{(s)}\). Hence \(W^{(s)}W^{(t)}\) and \(W^{(s+t)}\) are both intertwiners from \(H^{(s+t)}\) to \(H\), and \(W^{(t)*}\), \(W^{(-t)}\) are both intertwiners from \(H\) to \(H^{(t)}\). Since \(S^{is}S^{it}=S^{i(s+t)}\) and \((S^{it})^*=S^{-it}\), these pairs agree \(\Lambda_\nu\)-almost everywhere, hence off a \(\Lambda\)-negligible saturated set by Lemma 3.4 (which applies to bounded intertwiners of any twisted representations, with the same proof). Discarding countably many sets, for every \(x\) the map \(t\mapsto W^{(t)}_x\) on \(\mathbb Q\) is a group homomorphism into partial isometries with \(W^{(t)*}_x=W^{(-t)}_x\); so \(e_x=W^{(0)}_x\) is a projection and \(t\mapsto W^{(t)}_x\) is a unitary group of \(e_xH_x\).

Let \(C\) be the set of \(x\) where \(t\mapsto W^{(t)}_x\) is strongly continuous on \(\mathbb Q\) (uniformly on bounded sets). Since the \(W^{(t)}_x\) are contractions, it suffices to test on a countable family of sections dense in every fibre, so \(C\) is Borel. From \(W^{(t)}_y=\delta(\gamma)^{-it}U(\gamma)W^{(t)}_xU(\gamma)^*\) we see that \(C\) is saturated. The operators \(S^{it}=\nu(W^{(t)})\), \(t\in\mathbb Q\), are decomposable, hence so are their strong limits \(S^{it}\), \(t\in\mathbb R\); by (B5), \(S=\int^\oplus S_x\,d\Lambda_\nu(x)\) for a measurable field of positive self-adjoint operators \(S_x\), and \(W^{(t)}_x=S_x^{it}\) for all rational \(t\) and \(\Lambda_\nu\)-almost every \(x\). Such \(x\) lie in \(C\). So \(C\) has \(\Lambda\)-negligible complement (R3), and we discard it.

For \(x\in C\), the group extends by continuity to a strongly continuous unitary group of \(e_xH_x\); by Stone's theorem it is \(A_x^{it}\) for a unique nonsingular positive self-adjoint operator \(A_x\) on \(e_xH_x\). Put \(T_x=A_x\oplus0\) on \(e_xH_x\oplus(1-e_x)H_x\), so that \(T_x^{it}=W^{(t)}_x\). Measurability: the self-adjoint operators \(L_x=\log A_x\oplus0\) have \(e^{itL_x}=W^{(t)}_x+1-e_x\), a measurable field for rational \(t\), so \((L_x)\) is a measurable field by Unbounded decomposable operators and measurable spectral decomposition, Proposition 3.2(d); hence so is \((T_x)\), whose Cayley transform is \((c\circ\exp)(L_x)\,e_x-(1-e_x)\) with \(c(\lambda)=(\lambda-i)(\lambda+i)^{-1}\). The intertwining relation of Lemma 3.2(c) holds for rational \(t\), hence for all \(t\) by continuity, so \(T\) has degree \(1\). Finally \(\nu(T)^{it}=S^{it}\) for rational \(t\), hence for all \(t\), so \(\nu(T)=S\) by Stone's theorem. \(\square\)

4. Normal weights determined by a generating cone

The weight \(\varphi_T\) will be defined by its values on the operators \(\theta_\nu(\xi,\xi)\). The following general principle shows that such values determine a normal weight.

Lemma 4.1. Fix a von Neumann algebra \(M\) and a set \(\mathcal S\subset M_+\) with (i) \(x+y\in\mathcal S\) for \(x,y\in\mathcal S\), (ii) \(axa^*\in\mathcal S\) for \(x\in\mathcal S\), \(a\in M\), and (iii) the linear span of \(\mathcal S\) weakly dense in \(M\). Then \(\mathcal S_1=\{x\in\mathcal S:\|x\|<1\}\) is upward directed, and as an increasing net it converges strongly to \(1\).

Proof. By (ii) with \(a=\sqrt c\), \(\mathcal S\) is stable under multiplication by \(c>0\). For \(x\in\mathcal S_1\) put \(x'=x(1-x)^{-1}=(1-x)^{-1/2}x(1-x)^{-1/2}\in\mathcal S\) by (ii). Note that \(1+x'=(1-x)^{-1}\), so \(x'(1+x')^{-1}=x\). Given \(a,b\in\mathcal S_1\), let \(c=(a'+b')(1+a'+b')^{-1}=(1+a'+b')^{-1/2}(a'+b') (1+a'+b')^{-1/2}\). It lies in \(\mathcal S\) by (i) and (ii), and \(\|c\|<1\). The function \(t\mapsto t(1+t)^{-1}= 1-(1+t)^{-1}\) is operator monotone on \([0,\infty)\), because \(u\mapsto u^{-1}\) reverses order on positive invertible operators. Hence \(c\ge a'(1+a')^{-1}=a\) and likewise \(c\ge b\). So \(\mathcal S_1\) is upward directed; let \(e\le1\) be the strong limit of the increasing net.

For \(x\in\mathcal S\) and \(n\ge1\), \(nx(1+nx)^{-1}\in\mathcal S_1\), and these operators increase strongly to the support projection \(s(x)\). Hence \(e\ge s(x)\) for every \(x\in\mathcal S\). Let \(p=1-\bigvee_{x\in\mathcal S}s(x)\). Then \(xp=0\) for all \(x\in\mathcal S\), hence \(yp=0\) for all \(y\) in the span of \(\mathcal S\), and by weak density \(p=0\). So \(e\ge1\). \(\square\)

Proposition 4.2. Let \(\mathcal S\) be as in Lemma 4.1 and \(\varphi_1,\varphi_2\) normal weights on \(M\). If \(\varphi_1(x)\le\varphi_2(x)\) for all \(x\in\mathcal S\), then \(\varphi_1\le\varphi_2\). In particular normal weights that agree on \(\mathcal S\) are equal.

Proof. Let \(x\in M_+\) and let \((s_i)\) be the increasing net \(\mathcal S_1\). Then \(x^{1/2}s_ix^{1/2}\in \mathcal S\) by (ii), and it increases strongly to \(x\). By normality \(\varphi_1(x)=\sup_i\varphi_1(x^{1/2}s_i x^{1/2})\le\sup_i\varphi_2(x^{1/2}s_ix^{1/2})=\varphi_2(x)\). \(\square\)

Corollary 4.3. Let \(H\) be a \(\Lambda\)-random Hilbert space and \(\nu\in\mathcal E^+\) faithful. The set \(\mathcal S_\nu\) of finite sums \(\sum_i\theta_\nu(\xi_i,\xi_i)\), \(\xi_i\in D(U,\nu)\), satisfies the hypotheses of Lemma 4.1 in \(M=\operatorname{End}_\Lambda(H)\). Hence a normal weight on \(\operatorname{End}_\Lambda(H)\) is determined by its values on the operators \(\theta_\nu(\xi,\xi)\), \(\xi\in D(U,\nu)\).

Proof. (i) is clear. (ii): \(a\theta_\nu(\xi,\xi)a^*=\theta_\nu(a\xi,a\xi)\) with \(a\xi\in D(U,\nu)\) by (1.2) and (Q1). (iii): by polarization, \(\theta_\nu(\xi,\eta)=\frac14\sum_{k=0}^3i^k\theta_\nu(\xi+i^k\eta,\xi+i^k\eta)\), so the span of \(\mathcal S_\nu\) is \(\mathcal J_\nu\), which is weakly dense by (Q6). \(\square\)

5. The weight of a positive operator of degree one

Throughout this section \(H\) is a \(\Lambda\)-random Hilbert space and \(T\) a positive random operator of degree \(1\) on \(H\).

Lemma 5.1. Let \(\nu\in\mathcal E^+\), \(\psi=\psi_\nu\), and \(\xi\in D(U,\nu)\) with \(\int\|\xi_x\|^2 d\Lambda_\nu(x)<\infty\), and let \(\xi'\) be its class in \(\nu(H)\). Then \(\xi'\) is \(\psi\)-bounded in the \(W(\nu)\)-module \(\nu(H)\), \(R^\psi(\xi')=\nu(T_\nu(\xi)^*)\), and \[ \theta^\psi(\xi',\xi')=\nu\big(\theta_\nu(\xi,\xi)\big). \]

Proof. By (Q4) and (B10)(i), \(\nu(L^\nu)=L^2(G,m_\nu)\) is the GNS space of \(\psi\), with \(\eta_\psi(\lambda(f))=f\) for \(f\in\mathcal A_\nu\). The operator \(R=\nu(T_\nu(\xi)^*)\) from \(L^2(G,m_\nu)\) to \(\nu(H)\) is bounded (by \(\sup_y\|T_\nu(\xi)_y\|\)) and \(W(\nu)\)-linear, because \(T_\nu(\xi)^*\) is an intertwiner (Q4). For \(f\in\mathcal A_\nu\) and almost every \(y\), \(f|_{G^y}\) lies in \(L^1\cap L^2(G^y,\nu^y)\), so \(T_\nu(\xi)_y^*f=\int f(\gamma)U(\gamma)\xi_{s(\gamma)}\,d\nu^y(\gamma)\) (Q1). Since \(\xi'\) is bounded and square integrable, the description of the action in (Q4) gives \[ R\,\eta_\psi(\lambda(f))=Rf=\big(U(f\nu)\xi\big)'=\lambda(f)\xi'\qquad(f\in\mathcal A_\nu). \] We show \(y\xi'=R\eta_\psi(y)\) for every \(y\in\mathfrak n_\psi\). Proved in Weights and the Hilbert spaces of multiplication, §§WH-04–WH-08 and WH-11: the canonical weight and its GNS space in WH-05–WH-08, the left-bounded vectors \(\eta_\psi(\mathfrak n_\psi)\) in WH-11, and \(\lambda_\xi\zeta=\pi_r(\zeta)\xi\) in WH-04. Let \(V:\nu(H)\to\ell^2\otimes L^2(G,m_\nu)\) be an isometry with \(V\lambda(f)=(1\otimes\lambda(f))V\) (Q4), and write \(V\xi'=(\zeta_k)_k\) and \(VR=(R_k)_k\), with \(\zeta_k\in L^2(G,m_\nu)\) and bounded operators \(R_k\) on \(L^2(G,m_\nu)\). Applying \(V\) to this identity and taking the \(k\)-th component, \(\lambda(a)\zeta_k=R_ka\) for \(a\in\mathcal A_\nu\). So \(\zeta_k\) is right bounded in the sense of (B10)(ii), with \(\pi_r(\zeta_k)=R_k\), and \(y\zeta_k=R_k\eta_\psi(y)\) for \(y\in\mathfrak n_\psi\). Hence \(Vy\xi'=(1\otimes y)V\xi'=VR\eta_\psi(y)\), and \(y\xi'=R\eta_\psi(y)\) because \(V\) is isometric. Thus \(\|y\xi'\|\le\|R\|\|\eta_\psi(y)\|\), \(\xi'\) is \(\psi\)-bounded, and \(R^\psi(\xi')=R\). Finally \(\theta^\psi(\xi',\xi')=RR^*=\nu(T_\nu(\xi)^*T_\nu(\xi))=\nu(\theta_\nu(\xi,\xi))\). \(\square\)

Lemma 5.2. For every \(\nu\in\mathcal E^+\), the positive self-adjoint operator \(\nu(T)=\int^\oplus T_x\, d\Lambda_\nu(x)\) on \(\nu(H)\) is homogeneous of degree \(-1\) with respect to \(\psi_\nu\): \(\nu(T)^{it}\sigma^{\psi_\nu}_t(y)=y\,\nu(T)^{it}\) for \(y\in W(\nu)\), \(t\in\mathbb R\). Equivalently, \(\nu(T)^{it}\) is \(W(\nu)\)-linear from \(\nu(H^{(t)})\) to \(\nu(H)\).

Proof. Both sides are \(\sigma\)-weakly continuous and linear in \(y\), and \(\lambda(\mathcal A_\nu)\) is a \(*\)-algebra generating \(W(\nu)\), so it suffices to take \(y=\lambda(f)\), \(f\in\mathcal A_\nu\). By (1.3), \(\sigma_t(\lambda(f))=\lambda(\delta^{it}f)\), which acts on \(\nu(H)\) as \(U(\delta^{it}f\nu)\). For a section \(\zeta\) of \(\nu(H)\), Lemma 3.2(c) gives \[ \big(\nu(T)^{it}U(\delta^{it}f\nu)\zeta\big)_y=\int f(\gamma)\,T_y^{it}\delta(\gamma)^{it}U(\gamma)\zeta_{s(\gamma)}\, d\nu^y(\gamma)=\int f(\gamma)\,U(\gamma)T^{it}_{s(\gamma)}\zeta_{s(\gamma)}\,d\nu^y(\gamma)=\big(U(f\nu)\nu(T)^{it} \zeta\big)_y, \] where the bounded operator \(T^{it}_y\) passes through the weak integral. The second statement is the first one rewritten with the description of \(\nu(H^{(t)})\) before Lemma 3.5, because two normal representations of \(W(\nu)\) that agree on \(\lambda(\mathcal A_\nu)\) agree. \(\square\)

Theorem 5.3. Let \(H\) be a \(\Lambda\)-random Hilbert space.

(a) For every positive random operator \(T\) of degree \(1\) on \(H\) there is a unique normal semifinite weight \(\varphi_T\) on \(\operatorname{End}_\Lambda(H)\) satisfying \[ \varphi_T\big(\theta_\nu(\xi,\xi)\big)=\int\langle T_x\xi_x,\xi_x\rangle\,d\Lambda_\nu(x)\qquad(\nu\in\mathcal E^+,\ \xi\in D(U,\nu)). \tag{5.1} \] For every faithful \(\nu\), \(\nu(T)\) is the spatial derivative of the weight \(\varphi_T\circ\nu^{-1}\) on \(W(\nu)'\) with respect to \(\psi_\nu\).

(b) The map \(T\mapsto\varphi_T\) is a bijection from positive random operators of degree \(1\) onto normal semifinite weights on \(\operatorname{End}_\Lambda(H)\). The weight \(\varphi_T\) is faithful iff \(T_x\) is nonsingular for \(\Lambda\)-almost every \(x\).

Proof. Step 1: one faithful \(\nu\). Fix \(\nu\) faithful. By Lemma 5.2 and (S4) there is a unique normal semifinite weight \(\varphi^\nu\) on \(W(\nu)'\) with \(d\varphi^\nu/d\psi_\nu=\nu(T)\). Put \(\Phi_\nu=\varphi^\nu \circ\nu\), a normal semifinite weight on \(\operatorname{End}_\Lambda(H)\) by (Q5). If \(\xi\in D(U,\nu)\) and \(\int\|\xi_x\|^2d\Lambda_\nu<\infty\), then by Lemma 5.1 and (S2) \[ \Phi_\nu(\theta_\nu(\xi,\xi))=\varphi^\nu(\theta^\psi(\xi',\xi'))=\langle\nu(T)\xi',\xi'\rangle=\int\langle T_x\xi_x, \xi_x\rangle\,d\Lambda_\nu(x); \] the last equality is the description of the form of a direct integral: \(\xi'\in\operatorname{dom}\nu(T)^{1/2}\) iff \(\xi_x\in\operatorname{dom}T_x^{1/2}\) almost everywhere and \(\int\|T_x^{1/2}\xi_x\|^2<\infty\).

For general \(\xi\in D(U,\nu)\), choose Borel sets \(Z_n\uparrow G^{(0)}\) with \(\Lambda_\nu(Z_n)<\infty\) (standing assumption) and put \(\xi^n=1_{Z_n}\xi\). Then \(\xi^n\in D(U,\nu)\) with the same constant, and by (1.2) \(\theta_\nu(\xi^n,\xi^n)_y=\int1_{Z_n}(s(\gamma))(U(\gamma)\xi_{s(\gamma)})(U(\gamma)\xi_{s(\gamma)})^*d\nu^y(\gamma)\) increases to \(\theta_\nu(\xi,\xi)_y\) for every \(y\). So \(\theta_\nu(\xi^n,\xi^n)\uparrow\theta_\nu(\xi,\xi)\) in \(\operatorname{End}_\Lambda(H)\), and normality and monotone convergence give (5.1) for \(\xi\) and \(\nu\).

Step 2: a comparison. Let \(\nu_1\) be faithful and \(\nu_2=(g\circ s)\nu_1\) with \(0\le g\le1\). For \(\xi\in D(U,\nu_2)\), the section \(g^{1/2}\xi\) lies in \(D(U,\nu_1)\), since \(\int|\langle\alpha,U(\gamma)g(s(\gamma))^{1/2}\xi_{s(\gamma)}\rangle|^2d\nu_1^y=\int|\langle\alpha,U(\gamma) \xi_{s(\gamma)}\rangle|^2d\nu_2^y\), and (1.2) gives \(\theta_{\nu_1}(g^{1/2}\xi,g^{1/2}\xi)=\theta_{\nu_2}(\xi,\xi)\). Using \(\Lambda_{\nu_2}=g\Lambda_{\nu_1}\) (R2) and Step 1 for \(\nu_1\), \[ \Phi_{\nu_1}(\theta_{\nu_2}(\xi,\xi))=\int g(x)\langle T_x\xi_x,\xi_x\rangle\,d\Lambda_{\nu_1}(x)=\int\langle T_x\xi_x,\xi_x\rangle\,d\Lambda_{\nu_2}(x). \]

Step 3: independence. Let \(\nu\) be faithful and \(\nu'\in\mathcal E^+\) arbitrary. Then \(\nu_1=\nu+\nu'\) is faithful and \(\nu=(g\circ s)\nu_1\), \(\nu'=(g'\circ s)\nu_1\) with \(0\le g,g'\le1\) by (R1). By Step 2, \(\Phi_{\nu_1}\) satisfies (5.1) on the operators \(\theta_\nu(\xi,\xi)\); so does \(\Phi_\nu\), by Step 1. By Corollary 4.3, \(\Phi_{\nu_1}=\Phi_\nu\). By Step 2 again, \(\Phi_{\nu_1}\) satisfies (5.1) on the operators \(\theta_{\nu'}(\xi,\xi)\). Hence \(\varphi_T:=\Phi_\nu\) does not depend on the faithful \(\nu\) and satisfies (5.1) for all \(\nu'\). Uniqueness follows from Corollary 4.3, and the last assertion of (a) holds by construction.

Step 4: (b). Fix \(\nu\) faithful. If \(\varphi_{T_1}=\varphi_{T_2}\), then by (a) and (S2) \(\nu(T_1)=\nu(T_2)\), so \(T_{1,x}=T_{2,x}\) for \(\Lambda_\nu\)-almost every \(x\), and \(T_1=T_2\) by Lemma 3.4. Let \(\varphi\) be a normal semifinite weight on \(\operatorname{End}_\Lambda(H)\) and \(S=d(\varphi\circ\nu^{-1})/d\psi_\nu\). By (S4), \(S\) is homogeneous of degree \(-1\), which by the argument of Lemma 5.2 means that \(S^{it}\) is \(W(\nu)\)-linear from \(\nu(H^{(t)})\) to \(\nu(H)\). Lemma 3.5 gives a positive random operator \(T\) of degree \(1\) with \(\nu(T)=S\); then \(\varphi_T\circ\nu^{-1}\) and \(\varphi\circ\nu^{-1}\) have the same spatial derivative, so \(\varphi_T=\varphi\) by (S2). Finally, by (S4) \(\varphi_T\) is faithful iff \(\nu(T)\) is nonsingular, that is, iff \(T_x\) is nonsingular for \(\Lambda_\nu\)-almost every \(x\); the set where \(T_x\) is singular is saturated by Lemma 3.2, so this is the same as \(\Lambda\)-almost everywhere. \(\square\)

Reference: [Connes 1979, Section VI, Lemma 3] asserts that every \(\xi\in D(U,\nu)\) defines a vector \(\xi'\) of \(\nu(H)\) with \(\theta^\psi(\xi',\xi')=\nu(\theta_\nu(\xi,\xi))\), and derives (5.1) from it. Such a vector exists only when \(\int\|\xi_x\|^2d\Lambda_\nu<\infty\) (for \(G=\mathbb R\) with only units, \(\nu^x=\varepsilon_x\), \(\Lambda_\nu\) Lebesgue measure, \(H_x=\mathbb C\) and \(\xi=1\), it does not), so Lemma 5.1 assumes this and Step 1 reaches every \(\xi\) by truncation.

Heuristically, (5.1) says \(\varphi_T(A)=\int\operatorname{Trace}(T_xA_x)\,d\Lambda(x)\): for \(A=\theta_\nu(\xi,\xi)\) the "integrand" is \(\operatorname{Trace}(T_yA_y)=\int\langle T_yU(\gamma)\xi_x,U(\gamma)\xi_x\rangle d\nu^y(\gamma)=\int\delta(\gamma)^{-1}\langle T_x\xi_x,\xi_x\rangle d\nu^y(\gamma)\) (with \(x=s(\gamma)\)), a sum over the orbit of \(y\), and (5.1) is what the modular relation (1.1) makes of "integrating it over the orbit space". We write \(\varphi_T(A)=\int\operatorname{Trace}(T_xA_x)\,d\Lambda(x)\) as a notation for \(\varphi_T(A)\), and \(\int\operatorname{Trace}(T_x)\,d\Lambda(x)\) for \(\varphi_T(1)\).

Corollary 5.4. Let \(T\), \(T_1\), \(T_2\) be positive random operators of degree \(1\), nonsingular almost everywhere, so that the weights \(\varphi_T\), \(\varphi_{T_j}\) are faithful. Then:

(a) \(\sigma^{\varphi_T}_t(A)=\big(T_x^{it}A_xT_x^{-it}\big)_x\) for \(A\in\operatorname{End}_\Lambda(H)\);

(b) \((D\varphi_{T_2}:D\varphi_{T_1})_t=\big(T_{2,x}^{it}T_{1,x}^{-it}\big)_x\).

Proof. The families on the right have degree \(0\) by Lemma 3.2(b), since the factors \(\delta(\gamma)^{\pm it}\) cancel; they are bounded, so they are elements of \(\operatorname{End}_\Lambda(H)\). Fix \(\nu\) faithful. By Theorem 5.3(a), \(\nu(T)=d(\varphi_T\circ\nu^{-1})/d\psi_\nu\), and (S3) gives \(\nu(\sigma_t^{\varphi_T}(A))=\nu(T)^{it}\nu(A) \nu(T)^{-it}\), which is the image under \(\nu\) of the family in (a). Likewise (S3) gives \(\nu((D\varphi_{T_2}: D\varphi_{T_1})_t)=\nu(T_2)^{it}\nu(T_1)^{-it}\). Since \(\nu\) is injective (Q5), (a) and (b) follow. \(\square\)

Proposition 5.5 (transport). (a) Let \(V\in\operatorname{Hom}_\Lambda(H_1,H_2)\) be an isometry and \(T_1\) a positive random operator of degree \(1\) on \(H_1\). Then \(T_2=VT_1V^*\), which is \(T_1\) transported to \(VV^*H_2\) and \(0\) on \((1-VV^*)H_2\), is a positive random operator of degree \(1\) on \(H_2\), and \(\varphi_{T_2}(X)=\varphi_{T_1}(V^*XV)\) for \(X\in\operatorname{End}_\Lambda(H_2)_+\).

(b) Let \(H=\bigoplus_iH_i\) (countable sum), \(V_i\) the inclusions and \(T=\bigoplus_iT_i\). Then \(\varphi_T(X)=\sum_i\varphi_{T_i}(V_i^*XV_i)\) for \(X\in\operatorname{End}_\Lambda(H)_+\).

Proof. (a) Degree \(1\) is clear from (3.1) since \(V\) has degree \(0\). Both sides are normal weights in \(X\). For \(\xi\in D(U_2,\nu)\), \(V^*\theta_\nu(\xi,\xi)V=\theta_\nu(V^*\xi,V^*\xi)\) by (1.2), and \(\langle T_2\xi,\xi \rangle=\|T_1^{1/2}V^*\xi\|^2=\langle T_1V^*\xi,V^*\xi\rangle\) fibrewise. So both weights take the value \(\int\langle T_{1,x}V_x^*\xi_x,V_x^*\xi_x\rangle d\Lambda_\nu\) on \(\theta_\nu(\xi,\xi)\), and Corollary 4.3 gives equality. (b) Both sides are normal weights, and on \(\theta_\nu(\xi,\xi)\) both equal \(\sum_i\int\langle T_{i,x}V_{i,x}^*\xi_x,V_{i,x}^*\xi_x\rangle d\Lambda_\nu\), because the quadratic form of a direct sum is the sum of the quadratic forms. \(\square\)

Example 5.6 (one orbit). Let \(X\) be a standard Borel space with a \(\sigma\)-finite measure \(\beta\neq0\) and \(G=X\times X\) the pair groupoid, \(r(y,x)=y\), \(s(y,x)=x\), \(\delta=1\). A transverse function is \(\nu_\beta^y=\varepsilon_y\times\beta\) (the measure \(\beta\) placed on \(G^y=\{y\}\times X\)); every proper transverse function has this form. Transverse measures are \(\Lambda(\nu_\beta)=c\,\beta(X)\), \(c\ge0\): indeed, \(\Lambda_{\nu_\beta}\) must be a multiple \(c_\beta\beta\) for \(m_{\nu_\beta}=\Lambda_{\nu_\beta}\times\beta\) to be invariant under the flip, and (1.1), applied to the indicator function of a rectangle \(A\times A'\) with \(0<\beta(A')<\infty\) and \(0<\beta'(A)<\infty\), gives \(c_\beta\beta(A')\beta'(A)=c_{\beta'}\beta'(A)\beta(A')\), so \(c_\beta=c_{\beta'}\). Using \(U(x_0,x)\) to identify every \(H_x\) with \(K=H_{x_0}\), a section is a function \(\xi:X\to K\) and \(\theta_{\nu_\beta}(\xi,\xi)=\int_X\xi(x)\xi(x)^*d\beta(x)\) in \(\operatorname{End}_\Lambda(H)=\mathcal L(K)\) (we take \(c>0\); intertwiners are constant families under this identification). A positive operator of degree \(1\) is a single positive operator \(T_0\) on \(K\), and (5.1) reads \[ \varphi_T\Big(\int\xi(x)\xi(x)^*d\beta(x)\Big)=c\int\langle T_0\xi(x),\xi(x)\rangle d\beta(x), \] so \(\varphi_T=c\operatorname{Tr}(T_0^{1/2}\,\cdot\,T_0^{1/2})\). The orbit space is one point carrying mass \(c\).

6. Averaging along orbits and the operator-valued weight \(E_\nu\)

For a Hilbert space \(K\) and a positive self-adjoint \(S\) on \(K\), put \(S_\varepsilon=S(1+\varepsilon S)^{-1}\) (bounded) and \[ \omega_S(B)=\sup_{\varepsilon>0}\operatorname{Tr}\big(S_\varepsilon^{1/2}BS_\varepsilon^{1/2}\big)\qquad(B\in \mathcal L(K)_+). \] This is the usual weight "\(\operatorname{Trace}(S\,\cdot\,)\)".

Lemma 6.1. \(\omega_S\) is a normal weight on \(\mathcal L(K)\), and for every strongly convergent sum \(B=\sum_j\eta_j\eta_j^*\), \[ \omega_S(B)=\sum_j\langle S\eta_j,\eta_j\rangle . \] In particular \(\omega_S(X^*X)=\sum_k\langle SX^*e_k,X^*e_k\rangle\) for an orthonormal basis \((e_k)\), which is \(\|XS^{1/2}\|_{HS}^2\) when \(S\) is bounded.

Proof. The function \(t\mapsto t(1+\varepsilon t)^{-1}\) increases as \(\varepsilon\downarrow0\), so \(S_\varepsilon\) increases, and \(\operatorname{Tr}(S_\varepsilon^{1/2}\eta\eta^*S_\varepsilon^{1/2})=\langle S_\varepsilon\eta,\eta \rangle=\int t(1+\varepsilon t)^{-1}\,d\mu_\eta(t)\) increases to \(\int t\,d\mu_\eta(t)=\langle S\eta,\eta\rangle\) by monotone convergence (\(\mu_\eta\) the spectral measure of \(\eta\)). Each \(B\mapsto\operatorname{Tr}(S_\varepsilon^{1/2} BS_\varepsilon^{1/2})\) is a normal weight, and an increasing limit of normal weights is a normal weight (two suprema commute). By normality \(\operatorname{Tr}(S_\varepsilon^{1/2}BS_\varepsilon^{1/2})=\sum_j\langle S_\varepsilon\eta_j, \eta_j\rangle\), and taking the supremum over \(\varepsilon\) (monotone in both indices) gives the formula. For \(B=X^*X=\sum_k(X^*e_k)(X^*e_k)^*\) we get the second statement. \(\square\)

Let \(\nu\in\mathcal E^+\). Let \(P_\nu=\int^\oplus\mathcal L(H_x)\,d\Lambda_\nu(x)\), the von Neumann algebra of decomposable operators on \(\nu(H)\); it is of type I, and \(\nu\) maps \(\operatorname{End}_\Lambda(H)\) into it. For a positive random operator \(T\) of degree \(1\) put \[ \rho^\nu_T(B)=\int\omega_{T_x}(B_x)\,d\Lambda_\nu(x)\qquad(B\in(P_\nu)_+). \tag{6.1} \] The integrand is Borel: with a measurable orthonormal frame \((\xi^k)\) it is \(\sup_n\sum_k\langle B_xT_{1/n,x}^{1/2} \xi^k_x,T_{1/n,x}^{1/2}\xi^k_x\rangle\). Each functional \(B\mapsto\int_Z\langle B_xa_x,a_x\rangle d\Lambda_\nu\) with \(\Lambda_\nu(Z)<\infty\) and \(a\) bounded is a normal vector functional, and \(\rho^\nu_T\) is an increasing limit of countable sums of such functionals; so \(\rho^\nu_T\) is a normal weight. If \(\tau_\nu=\int\operatorname{Tr}\, d\Lambda_\nu\) denotes the canonical fns trace of \(P_\nu\) and \(T\) is nonsingular, then \(\rho^\nu_T\) is the weight \(\tau_h\) of (B3) with \(h=\nu(T)\); hence it is fns, \(\sigma^{\rho^\nu_T}_t=\operatorname{Ad}\nu(T)^{it}\), and \((D\rho^\nu_{T_2}:D\rho^\nu_{T_1})_t=\nu(T_2)^{it}\nu(T_1)^{-it}\).

Proposition 6.2 (averaging formula). Let \(\nu\in\mathcal E^+\) and let \(B=(B_x)\) be a bounded Borel family of positive operators such that the family \[ C_y=\int_{G^y}U(\gamma)B_{s(\gamma)}U(\gamma)^*\,d\nu^y(\gamma)\qquad(\text{weak integral}) \tag{6.2} \] is bounded. Then:

(a) \(C\) is an intertwiner, \(C\in\operatorname{End}_G(H)\);

(b) for every bounded Borel section \(\xi\), the section \(B^{1/2}\xi\) lies in \(D(U,\nu)\);

(c) \(C=\sum_j\theta_\nu(B^{1/2}\xi^j,B^{1/2}\xi^j)\) for every measurable orthonormal frame \((\xi^j)\) of \(H\);

(d) \(\rho^\nu_T(B)=\varphi_T(C)\) for every positive random operator \(T\) of degree \(1\).

Proof. (a) For \(\gamma_0:x\to y\), left invariance \(\gamma_0\nu^x=\nu^y\) gives \(U(\gamma_0)C_xU(\gamma_0)^*= \int U(\gamma_0\gamma)B_{s(\gamma)}U(\gamma_0\gamma)^*d\nu^x(\gamma)=C_y\). Measurability is clear.

(b) For \(\alpha\in H_y\), by the Cauchy–Schwarz inequality \(|\langle\alpha,U(\gamma)B_x^{1/2}\xi_x\rangle|^2= |\langle B_x^{1/2}U(\gamma)^*\alpha,\xi_x\rangle|^2\le\|\xi\|_\infty^2\langle U(\gamma)B_xU(\gamma)^*\alpha,\alpha\rangle\). Integrating over \(\nu^y\) gives the bound \(\|\xi\|_\infty^2\|C\|\|\alpha\|^2\).

(c) With \(\eta^j=B^{1/2}\xi^j\) we have \(\sum_j\eta^j_x\eta^{j*}_x=B_x^{1/2}\big(\sum_j\xi^j_x\xi^{j*}_x\big) B_x^{1/2}=B_x\), so by (1.2) and monotone convergence, \(\sum_j\theta_\nu(\eta^j,\eta^j)_y=\int U(\gamma) \big(\sum_j\eta^j\eta^{j*}\big)_{s(\gamma)}U(\gamma)^*d\nu^y(\gamma)=C_y\).

(d) By (c), normality, (5.1), monotone convergence and Lemma 6.1, \[ \varphi_T(C)=\sum_j\int\langle T_x\eta^j_x,\eta^j_x\rangle\,d\Lambda_\nu(x)=\int\sum_j\langle T_x\eta^j_x,\eta^j_x\rangle \,d\Lambda_\nu(x)=\int\omega_{T_x}(B_x)\,d\Lambda_\nu(x).\qquad\square \]

Theorem 6.3. Let \(\nu\in\mathcal E^+\) be faithful, and identify \(\operatorname{End}_\Lambda(H)\) with its image under \(\nu\) in \(P_\nu\). There is a unique operator-valued weight \(E_\nu\) from \(P_\nu\) to \(\operatorname{End}_\Lambda(H)\) with \(\varphi_T\circ E_\nu=\rho^\nu_T\) for one (equivalently, for every) nonsingular positive random operator \(T\) of degree \(1\). It is faithful, normal and semifinite, and:

(a) \(\varphi_T\circ E_\nu=\rho^\nu_T\) for every positive random operator \(T\) of degree \(1\), singular or not;

(b) \(E_\nu(B)=C\) whenever \(B\in(P_\nu)_+\) and the average \(C\) of (6.2) is bounded; in particular \(E_\nu(\eta\eta^*)=\theta_\nu(\eta,\eta)\) for \(\eta\in D(U,\nu)\).

Proof. Existence. \(\operatorname{End}_\Lambda(H)\) has a faithful normal state, which is \(\varphi_{T_0}\) for some nonsingular \(T_0\) by Theorem 5.3(b). The fns weight \(\rho^\nu_{T_0}\) has modular group \(\operatorname{Ad} \nu(T_0)^{it}\), whose restriction to \(\operatorname{End}_\Lambda(H)\) is \(\sigma^{\varphi_{T_0}}\) (Corollary 5.4). By (B1) there is a unique fns operator-valued weight \(E_\nu\) with \(\varphi_{T_0}\circ E_\nu=\rho^\nu_{T_0}\). For another nonsingular \(T_1\), (B3), Corollary 5.4(b) and (B1) give \[ (D\rho^\nu_{T_1}:D\rho^\nu_{T_0})_t=\nu(T_1)^{it}\nu(T_0)^{-it}=(D\varphi_{T_1}:D\varphi_{T_0})_t= (D(\varphi_{T_1}\circ E_\nu):D(\varphi_{T_0}\circ E_\nu))_t, \] so \(\varphi_{T_1}\circ E_\nu=\rho^\nu_{T_1}\) by (B2). Uniqueness is part of (B1).

(b), rank one. Let \(\eta\in D(U,\nu)\). For nonsingular \(T\), \(\varphi_T(E_\nu(\eta\eta^*))=\rho^\nu_T(\eta\eta^*)= \int\langle T_x\eta_x,\eta_x\rangle d\Lambda_\nu=\varphi_T(\theta_\nu(\eta,\eta))\) by (5.1). Let \(\omega\) be a normal positive functional on \(\operatorname{End}_\Lambda(H)\) and \(\omega_0\) a faithful normal state. The faithful normal positive functionals \(\omega_0\) and \(\omega+\omega_0\) are of the form \(\varphi_T\) with \(T\) nonsingular. Since \(\omega_0(E_\nu(\eta\eta^*))=\omega_0(\theta_\nu(\eta,\eta))<\infty\), subtraction gives \(\omega(E_\nu(\eta\eta^*))= \omega(\theta_\nu(\eta,\eta))\). An element of the extended positive part is determined by its values on normal positive functionals, so \(E_\nu(\eta\eta^*)=\theta_\nu(\eta,\eta)\).

(b), general. With \(\eta^j=B^{1/2}\xi^j\in D(U,\nu)\) as in Proposition 6.2, \(B=\sum_j\eta^j\eta^{j*}\), and normality of \(E_\nu\) and Proposition 6.2(c) give \(E_\nu(B)=\sum_j\theta_\nu(\eta^j,\eta^j)=C\).

(a). Let \(T\) be arbitrary and \(e=s(T)\in\operatorname{End}_\Lambda(H)\) (Lemma 3.2). The random Hilbert space \((1-e)H\) carries a nonsingular positive random operator \(R\) of degree \(1\) (Theorem 5.3(b) applied to a faithful normal state of \(\operatorname{End}_\Lambda((1-e)H)\)). Then \(T_1=T|_{eH}\oplus R\) is nonsingular of degree \(1\). By Proposition 5.5, \(\varphi_{T_1}(eXe)=\varphi_{T|_{eH}}(eXe|_{eH})=\varphi_T(X)\) for \(X\ge0\), and by normality this extends to the extended positive part. Fibrewise, \(\omega_{T_{1,x}}(e_xB_xe_x)=\omega_{T_x}(B_x)\): write \(B_x=\sum\eta_j\eta_j^*\) and use \(\langle T_{1}e\eta,e\eta\rangle=\langle T\eta,\eta\rangle\) (Lemma 6.1), so \(\rho^\nu_{T_1}(eBe)=\rho^\nu_T(B)\). Since \(E_\nu(eBe)=eE_\nu(B)e\) (B1), \[ \varphi_T(E_\nu(B))=\varphi_{T_1}(eE_\nu(B)e)=\varphi_{T_1}(E_\nu(eBe))=\rho^\nu_{T_1}(eBe)=\rho^\nu_T(B).\qquad\square \]

Example 6.4. \(E_\nu(1)\) is the average of the constant family \(1\), namely the family of scalars \((\nu^y(1)\,1_{H_y})_y\) (a function constant on orbits), provided it is bounded. So if \(\nu^y(1)=1\) for all \(y\), \(E_\nu\) is a faithful normal conditional expectation of \(P_\nu\) onto \(\operatorname{End}_\Lambda(H)\), given by \(E_\nu(B)_y=\int U(\gamma)B_{s(\gamma)}U(\gamma)^*d\nu^y(\gamma)\). In that case \(\operatorname{End}_\Lambda(H)\) is the range of a conditional expectation from a type I algebra.

Proposition 6.5. Let \(F\) be a proper random variable, \(H=L^2\circ F\), and for a Borel \(f\ge0\) on \(X=\bigsqcup F(x)\) let \(M(f)\) be the family of multiplications by \(f\). For every faithful \(\nu\) and bounded \(f\ge0\), \[ E_\nu(M(f))=M(\nu*f), \] where the right side is the element \(\sup_kM(\min(\nu*f,k))\) of the extended positive part; it is bounded iff \(\nu*f\) is essentially bounded for the measure \(\int\alpha^x\,d\Lambda_\nu(x)\) on \(X\), in particular when \(\nu*f\) is bounded.

Proof. The function \(\nu*f\) is \(G\)-invariant (R4), so \(M(\min(\nu*f,k))\in\operatorname{End}_\Lambda(H)\). As in Example 3.3(b), \(U(\gamma)M(f)_xU(\gamma)^*\) is multiplication by \(z\mapsto f(\gamma^{-1}z)\) on \(F(y)\), and by Fubini the weak integral over \(\nu^y\) is multiplication by \((\nu*f)(z)=\int f(\gamma^{-1}z)d\nu^y(\gamma)\). If \(\nu*f\) is bounded, Theorem 6.3(b) applies. In general, take \(f_1>0\) with \(\nu*f_1=1\) and \(g_k= \min(f,kf_1)\uparrow f\); then \(\nu*g_k\le k\), \(E_\nu(M(g_k))=M(\nu*g_k)\uparrow M(\nu*f)\), and normality of \(E_\nu\) concludes. For a bounded invariant \(g\ge0\), \(\nu(M(g))\) is multiplication by \(g\) on \(\nu(H)=\int^\oplus L^2(F(x),\alpha^x)\,d\Lambda_\nu(x)\), whose norm is the essential supremum of \(g\) for \(\int\alpha^x\,d\Lambda_\nu(x)\), and \(A\mapsto\nu(A)\) is isometric (Q3). So the increasing family \(M(\min(\nu*f,k))\) is bounded exactly when \(\nu*f\) is essentially bounded. \(\square\)

7. The modular spectrum and traces

A reference for the spectral theory used here is [Arveson 1974]. For a self-adjoint operator \(A\) write \(\operatorname{Sp}A\) for its spectrum.

Proposition 7.1. Let \(\mu\) be a \(\sigma\)-finite measure on a standard Borel space \(Y\), \((A_y)\) a measurable field of self-adjoint operators on \((K_y)\), \(A=\int^\oplus A_y\,d\mu\), \(P=\int^\oplus\mathcal L(K_y) \,d\mu\), and \(\alpha_t=\operatorname{Ad}e^{itA}\) on \(P\). For \(p\in\mathbb R\) and \(\varepsilon>0\) let \[ Y_\varepsilon(p)=\{y\in Y:(\operatorname{Sp}A_y-\operatorname{Sp}A_y)\cap(p-\varepsilon,p+\varepsilon)\neq\emptyset\}. \] Then \(Y_\varepsilon(p)\) is Borel, and \(p\in\operatorname{Sp}(\alpha)\) iff \(\mu(Y_\varepsilon(p))>0\) for every \(\varepsilon>0\).

Proof. Let \(e_y\) be the spectral measure of \(A_y\). For rationals \(a,b,r\) with \(r>0\) and \([a-b-2r,a-b+2r] \subset(p-\varepsilon,p+\varepsilon)\), let \(Y_{a,b,r}=\{y:e_y((a-r,a+r))\neq0\neq e_y((b-r,b+r))\}\), a Borel set. If \(y\in Y_{a,b,r}\), there are \(\lambda_1\in\operatorname{Sp}A_y\cap[a-r,a+r]\) and \(\lambda_2\in \operatorname{Sp}A_y\cap[b-r,b+r]\), so \(y\in Y_\varepsilon(p)\). Conversely, if \(\lambda_1-\lambda_2\in (p-\varepsilon,p+\varepsilon)\) with \(\lambda_j\) in the spectrum, choose a small rational \(r>0\) and rationals \(a,b\) within \(r\) of \(\lambda_1,\lambda_2\); then \(y\in Y_{a,b,r}\). So \(Y_\varepsilon(p)\) is the countable union of the \(Y_{a,b,r}\).

Suppose \(\mu(Y_\varepsilon(p))>0\) for all \(\varepsilon\). Fix \(\varepsilon\) and \((a,b,r)\) as above with \(\mu(Y_{a,b,r})>0\). Choosing, for \(y\in Y_{a,b,r}\), the first section \(\xi^n\) of a fixed countable dense family with \(e_y((a-r,a+r))\xi^n_y\neq0\), and normalizing, we get a Borel unit section \(u\) with \(u_y\in e_y((a-r,a+r))K_y\); likewise \(v\) for \((b-r,b+r)\). Let \(B_y=u_yv_y^*\) on \(Y_{a,b,r}\) and \(B_y=0\) elsewhere. Then \(B\neq0\), and with \(E_1=\int^\oplus e_y((a-r,a+r))\), \(E_2=\int^\oplus e_y((b-r,b+r))\), we have \(B=E_1BE_2\) and \(\alpha_t(B)= u_tBv_t^*\) where \(u_t=e^{itA}E_1\), \(v_t=e^{itA}E_2\) are unitary groups on the ranges of \(E_1\), \(E_2\), with spectra in \([a-r,a+r]\), \([b-r,b+r]\). By (B4)(iv), \(\operatorname{Sp}_\alpha(B)\subset[a-b-2r,a-b+2r]\subset (p-\varepsilon,p+\varepsilon)\); by (B4)(ii),(i) it is a nonempty subset of \(\operatorname{Sp}(\alpha)\). As \(\varepsilon\) is arbitrary and \(\operatorname{Sp}(\alpha)\) is closed, \(p\in\operatorname{Sp}(\alpha)\).

Suppose \(\mu(Y_\varepsilon(p))=0\) for some \(\varepsilon\). Choose \(f\in L^1(\mathbb R)\) (a Schwartz function) with \(\hat f(p)=1\) and \(\hat f\) supported in \([p-\varepsilon/2,p+\varepsilon/2]\). For \(B\in P\), \((\alpha(f)B)_y=\int f(t)e^{itA_y}B_ye^{-itA_y}dt\) for almost every \(y\). For \(y\notin Y_\varepsilon(p)\) the fibre action \(\alpha^y=\operatorname{Ad}e^{itA_y}\) has spectrum in the closure of \(\operatorname{Sp}A_y-\operatorname{Sp} A_y\) (B4)(iv), which does not meet the open interval \((p-\varepsilon,p+\varepsilon)\). So \(\hat f\) vanishes on a neighbourhood of \(\operatorname{Sp}_{\alpha^y}(B_y)\), and \(\alpha^y(f)B_y=0\) by (B4)(iii). Hence \(\alpha(f)=0\) while \(\hat f(p)\neq0\): \(p\notin\operatorname{Sp}(\alpha)\). \(\square\)

Theorem 7.2. Assume that \(G\) has trivial isotropy \(\Lambda\)-almost everywhere (Q7). Let \(T\) be a nonsingular positive random operator of degree \(1\) and \(\varphi=\varphi_T\). For \(p\in\mathbb R\) and \(\varepsilon>0\) the set \[ X_\varepsilon(p)=\{x\in G^{(0)}:(\operatorname{Sp}\log T_x-\operatorname{Sp}\log T_x)\cap(p-\varepsilon,p+\varepsilon) \neq\emptyset\} \] is saturated and Borel, and \[ p\in\operatorname{Sp}(\sigma^\varphi)\iff X_\varepsilon(p)\text{ is not }\Lambda\text{-negligible for any } \varepsilon>0. \] In multiplicative notation: \(e^p\) belongs to the modular spectrum of \(\varphi\) iff for every \(\varepsilon\) a non-negligible set of \(x\) has two spectral values of \(T_x\) with ratio in \((e^{p-\varepsilon},e^{p+\varepsilon})\).

Proof. By Lemma 3.2, \(\log T_y=U(\gamma)(\log T_x)U(\gamma)^*-\log\delta(\gamma)\), so \(\operatorname{Sp}\log T_y\) is a translate of \(\operatorname{Sp}\log T_x\) and \(X_\varepsilon(p)\) is saturated; it is Borel by Proposition 7.1. Fix a faithful \(\nu\) and put \(P=P_\nu\), \(\rho=\rho^\nu_T\), \(\mathcal M=\nu(\operatorname{End}_\Lambda(H))\subset P\). By Section 6, \(\sigma^\rho_t=\operatorname{Ad}\nu(T)^{it}\), and Proposition 7.1 with \(A_x=\log T_x\) and \(\mu=\Lambda_\nu\) (together with (R3)) shows that \(\operatorname{Sp}(\sigma^\rho)\) is the right-hand set. It remains to show \(\operatorname{Sp}(\sigma^\varphi)=\operatorname{Sp}(\sigma^\rho)\). By Corollary 5.4, \(\sigma^\rho|_{\mathcal M}=\sigma^\varphi\).

Let \(\mathcal D\) be the diagonal algebra of \(\nu(H)\). It commutes with every decomposable operator, in particular with \(\mathcal M\) and with \(\nu(T)^{it}\); so \(\sigma^\rho\) is trivial on \(\mathcal D\).

\(\mathcal M\) and \(\mathcal D\) generate \(P\). Let \(D\) be the countable set of (Q7) and \(b^{\xi,\eta}= \nu(\theta_\nu(\xi,\eta))\in\mathcal M\) for \(\xi,\eta\in D\). The points with nontrivial isotropy lie in a \(\Lambda\)-negligible set, so for \(\Lambda_\nu\)-almost every \(y\) the fibres \(b^{\xi,\eta}_y\) span a weakly dense \(*\)-subalgebra of \(\mathcal L(H_y)\). By (B5), the von Neumann algebra generated by \(\mathcal D\) and the \(b^{\xi,\eta}\) is \(\int^\oplus\mathcal L(H_y)\,d\Lambda_\nu(y)=P\). A fortiori \(\mathcal M\) and \(\mathcal D\) generate \(P\).

Equality of spectra. If \(\sigma^\rho(f)=0\) then \(\sigma^\varphi(f)=0\) by restriction. Conversely let \(\sigma^\varphi(f)=0\). For \(a\in\mathcal M\), \(b\in\mathcal D\), \(\sigma^\rho_t(ab)=\sigma^\varphi_t(a)b\), so \(\sigma^\rho(f)(ab)=(\sigma^\varphi(f)a)b=0\). The span of such products is a unital \(*\)-algebra (\(\mathcal M\) and \(\mathcal D\) commute), which is \(\sigma\)-weakly dense in \(P\) by the double commutant theorem, and \(\sigma^\rho(f)\) is \(\sigma\)-weakly continuous, so \(\sigma^\rho(f)=0\). The two actions have the same annihilating functions, hence the same spectrum. \(\square\)

The same argument works when the isotropy is nontrivial, provided that \(\delta=1\) on \(G^y_y\), that \(U(G^y_y)''\) is a factor, and that the fibres \(\theta_\nu(\xi,\eta)_y\) generate the commutant \(U(G^y_y)'\) for \(\Lambda\)-almost every \(y\): then the relative commutant \(\mathcal M'\cap P=\int^\oplus U(G^y_y)''\,d\Lambda_\nu\) replaces \(\mathcal D\). [Connes 1979, Section VI, Corollary 6] states the result under the first two conditions and obtains the third from its general generation statement for the isotropy commutant, which is not proved in this course (Remark 5.2 of "Square-integrable representations and random operators").

Corollary 7.3. Assume that \(G\) has trivial isotropy \(\Lambda\)-almost everywhere. A positive random operator \(T\) of degree \(1\) gives a trace \(\varphi_T\) iff \(T_x\) is a scalar \(\lambda_x1_{H_x}\) for \(\Lambda\)-almost every \(x\). The implication "scalar \(\Rightarrow\) trace" holds without the assumption on the isotropy.

Proof. Suppose \(T_x=\lambda_x1\). The set \(Y=\{\lambda_x>0\}\) is saturated (\(\lambda_y=\delta(\gamma)^{-1} \lambda_x\)) and \(e=s(T)=(1_Y(x)1)\) is central in \(\operatorname{End}_\Lambda(H)\), since scalar families commute with intertwiners. By (S2) and Theorem 5.3(a), \(s(\varphi_T)=e\), so \(\varphi_T(X)=\varphi_T(eXe)\) (B2). On \(e\operatorname{End}_\Lambda(H)e=\operatorname{End}_\Lambda(eH)\), \(\varphi_T\) is \(\varphi_{T|eH}\) (Proposition 5.5), which is faithful with modular group \(\operatorname{Ad}\lambda_x^{it}=\mathrm{id}\) (Corollary 5.4), hence a trace (B2). For \(A\in\operatorname{End}_\Lambda(H)\), \(eA=eAe\) and \(\varphi_T(AA^*)=\varphi_T(eAA^*e)=\varphi_T((eA)(eA)^*)=\varphi_T((eA)^*(eA))=\varphi_T(A^*A)\).

Conversely let \(\varphi_T\) be a trace. Its support \(e=s(T)\) is central (B2). By (Q7) the central elements are the invariant scalar families, so \(e=(1_Y(x)1)\) for a saturated Borel \(Y\), and \(T_x=0\) for almost every \(x\notin Y\). The restriction of \(\varphi_T\) to \(\operatorname{End}_\Lambda(eH)\) is a faithful trace, so its modular group is trivial and its spectrum is contained in \(\{0\}\) (B4)(v). Theorem 7.2, applied to \(eH\) (the zero fibres off \(Y\) play no role), shows that for each \(p\neq0\) some \(X_\varepsilon(p)\) is negligible. Covering \(\mathbb R\setminus\{0\}\) by countably many such intervals, we find that for almost every \(x\in Y\) the set \(\operatorname{Sp}\log T_x-\operatorname{Sp}\log T_x\) is \(\{0\}\), so \(\operatorname{Sp}\log T_x\) is one point and \(T_x\) is scalar. \(\square\)

Example 7.4. Some hypothesis on the isotropy is needed in Theorem 7.2 and Corollary 7.3. If \(G\) is a single compact group \(\Gamma\neq\{e\}\) (unit space a point, \(\delta=1\)) and \(H=L^2(\Gamma)\), the isotropy representation is the regular representation, which is not factorial, and \(\operatorname{End}_\Lambda(H)=R(\Gamma)\) is a direct sum of matrix algebras. A positive operator of degree \(1\) is a positive \(T\) commuting with \(\lambda(\Gamma)\), that is, \(T\in R(\Gamma)\) affiliated; taking \(T\) a non-scalar central projection plus \(1\) gives a non-scalar \(T\) for which \(\varphi_T\) is still a trace, since \(T\) is central in \(R(\Gamma)\) and \(\sigma^{\varphi_T}=\operatorname{Ad}T^{it}=\mathrm{id}\).

8. Integrable modular groups and the centralizer

A reference for integrable actions is [Connes–Takesaki 1977]. Let \(\alpha\) be a \(\sigma\)-weakly continuous action of \(\mathbb R\) on a von Neumann algebra \(M\). For \(x\in M_+\), \(E^\alpha(x)=\int_{\mathbb R} \alpha_t(x)\,dt\) is the element of the extended positive part of the fixed-point algebra \(M^\alpha\) given by \(\omega(E^\alpha(x))=\int\omega(\alpha_t(x))dt\). We call \(x\) integrable if \(E^\alpha(x)\) is bounded; the integrable elements form a hereditary cone \(\mathcal I_\alpha\) (if \(0\le y\le x\in\mathcal I_\alpha\) then \(y\in\mathcal I_\alpha\)).

Lemma 8.1. The following are equivalent; when they hold, \(\alpha\) is called integrable.

(1) \(\mathfrak n_\alpha=\{x\in M:x^*x\in\mathcal I_\alpha\}\) is \(\sigma\)-weakly dense in \(M\).

(2) \(\bigvee_{x\in\mathcal I_\alpha}s(x)=1\).

(3) There is an increasing net in \(\mathcal I_\alpha\) with supremum \(1\) (if \(M\) has separable predual, one can take a sequence).

Proof. (1)\(\Rightarrow\)(2): the \(\sigma\)-weak closure of the left ideal \(\mathfrak n_\alpha\) is \(Mq\) with \(q\) the join of the right supports \(r(x)=s(x^*x)\), \(x\in\mathfrak n_\alpha\); density forces \(q=1\). (2)\(\Rightarrow\)(3): the proof of Lemma 4.1 applies to \(\mathcal S=\mathcal I_\alpha\) with (ii) replaced by heredity, because \(x(1-x)^{-1}\le(1-\|x\|)^{-1}x\), \((a'+b')(1+a'+b')^{-1}\le a'+b'\) and \(nx(1+nx)^{-1}\le nx\), and with (iii) replaced by (2). If \(M\) has separable predual, it has a faithful normal state \(\omega\); choosing elements \(x_n\ge x_{n-1}\) of the net with \(\omega(1-x_n)<1/n\) gives a sequence with supremum \(1\). (3)\(\Rightarrow\)(1): if \(y_i\uparrow1\) in \(\mathcal I_\alpha\) and \(a\in M\), then \(ay_i^{1/2}\in\mathfrak n_\alpha\) because \(y_i^{1/2}a^*ay_i^{1/2}\le \|a\|^2y_i\), and \(ay_i^{1/2}\to a\) strongly. \(\square\)

For an fns weight \(\varphi\), the fixed-point algebra of \(\sigma^\varphi\) is the centralizer \(M_\varphi\), and we write \(E^\varphi=E^{\sigma^\varphi}\).

Lemma 8.2. Let \(N\subset M\) be von Neumann algebras, \(E\) an fns operator-valued weight from \(M\) to \(N\), \(\varphi\) an fns weight on \(N\) and \(\psi=\varphi\circ E\).

(a) If \(\sigma^\varphi\) is integrable, so is \(\sigma^\psi\).

(b) Suppose there is an increasing net \((x_i)\) in \(M_\psi\cap M_+\) with \(E(x_i)\) bounded and \(\sup_ix_i=1\). If \(\sigma^\psi\) is integrable, so is \(\sigma^\varphi\).

Proof. Recall from (B1) that \(\sigma^\psi_t|_N=\sigma^\varphi_t\) and \(E\circ\sigma^\psi_t=\sigma^\varphi_t\circ E\).

(a) Let \(y_j\uparrow1\) in \(\mathcal I_{\sigma^\varphi}\). Viewed in \(M\), \(E^\psi(y_j)=\int\sigma^\psi_t(y_j)dt= \int\sigma^\varphi_t(y_j)dt\) is bounded, so Lemma 8.1(3) holds for \(\sigma^\psi\).

(b) Let \(y\in\mathcal I_{\sigma^\psi}\), \(Y=E^\psi(y)\), and \(z_i=E(x_i^{1/2}yx_i^{1/2})\le\|y\|E(x_i)\), a bounded element of \(N_+\). Since \(x_i\) lies in the centralizer, \(\sigma^\psi_t(x_i^{1/2})=x_i^{1/2}\), and \[ \sigma^\varphi_t(z_i)=E(\sigma^\psi_t(x_i^{1/2}yx_i^{1/2}))=E(x_i^{1/2}\sigma^\psi_t(y)x_i^{1/2}). \] For \(\omega\in N_*^+\), \(\omega\circ E\) is a normal weight, hence a sum of normal positive functionals, and Tonelli's theorem gives \[ \int\omega(\sigma^\varphi_t(z_i))\,dt=(\omega\circ E)\Big(x_i^{1/2}\Big(\int\sigma^\psi_t(y)dt\Big)x_i^{1/2}\Big)\le \|Y\|\,\omega(E(x_i)). \] So \(E^\varphi(z_i)\le\|Y\|E(x_i)\) is bounded: \(z_i\in\mathcal I_{\sigma^\varphi}\). Now take \(y=y_j\) from an increasing net \(y_j\uparrow1\) in \(\mathcal I_{\sigma^\psi}\), and let \(p\in N\) be a projection orthogonal to all supports \(s(z_{i,j})\). Then \(pz_{i,j}p=0\); since \(x_i^{1/2}y_jx_i^{1/2}\uparrow x_i\), normality of \(E\) gives \(pE(x_i)p=\sup_jpz_{i,j}p=0\), so \(E(px_ip)=0\) and \(px_ip=0\) because \(E\) is faithful. Taking the supremum over \(i\), \(p=0\). Lemma 8.1(2) holds for \(\sigma^\varphi\). \(\square\)

Reference: [Connes 1979, Section VI, Lemma 13] states Lemma 8.2(b) for any increasing net in the domain of \(E\) inside the centralizer, without requiring \(\sup_ix_i=1\). This fails: take \(N=\mathbb C\subset M=B(L^2(\mathbb R))\), \(\varphi(\lambda)=\lambda\) and \(E=\omega_{e^Q}\) (Section 6), \(Q\) multiplication by the variable; then \(\psi=E\), the only such net is \(0\), \(\sigma^\psi\) is integrable (Proposition 8.3), and \(\sigma^\varphi\) is trivial, hence not integrable. So we assume \(\sup_ix_i=1\), which is what the applications provide.

We need the fibrewise criterion. Recall that for a vector \(a\) and a self-adjoint \(A\) with spectral measure \(e\), \(\mu_a(I)=\langle e(I)a,a\rangle\) and \(\mu_{a,\zeta}(I)=\langle e(I)a,\zeta\rangle\); then \(|\mu_{a,\zeta}(I)|^2\le\mu_a(I)\mu_\zeta(I)\). We use the following fact of harmonic analysis: a finite complex Borel measure \(\mu\) on \(\mathbb R\) has \(\hat\mu(t)=\int e^{it\lambda}d\mu(\lambda)\) square integrable iff \(\mu=g\,d\lambda\) with \(g\in L^2\), and then \(\int|\hat\mu|^2dt=2\pi\int|g|^2d\lambda\).

Proof of this fact. Write \(\check f(t)=\int f(\lambda)e^{it\lambda}\,d\lambda\) for \(f\in L^1(\mathbb R)\). For the pairing \(e^{it\lambda}\), the dual Haar measure of \(d\lambda\) is \(dt/2\pi\) (put \(t=2\pi\xi\) in The Fourier inversion theorem and the dual Haar measure, Proposition 3.2), and \(\check f\) differs from the Fourier transform there by the reflection \(t\mapsto-t\). So, by The Plancherel theorem, Theorem 1.1, \(f\mapsto\check f\) on \(L^1\cap L^2\) extends to a unitary operator \(\mathcal F\) of \(L^2(d\lambda)\) onto \(L^2(dt/2\pi)\). If \(\mu=g\,d\lambda\) with \(g\in L^2\), then \(g\in L^1\), because \(\mu\) is finite; so \(\hat\mu=\check g=\mathcal Fg\) and \(\int|\hat\mu|^2dt=2\pi\int|g|^2d\lambda\). Conversely, let \(\hat\mu\in L^2\) and put \(g=\mathcal F^{-1}\hat\mu\). For \(\varepsilon>0\) let \(G_\varepsilon(\lambda)=(4\pi\varepsilon)^{-1/2}e^{-\lambda^2/4\varepsilon}\), an even function with \(\check G_\varepsilon(t)=e^{-\varepsilon t^2}\). The function \((\mu*G_\varepsilon)(\lambda)=\int G_\varepsilon(\lambda-\kappa)\,d\mu(\kappa)\) lies in \(L^1\cap L^2\), with norms at most \(|\mu|(\mathbb R)\) times those of \(G_\varepsilon\), and Fubini's theorem gives \(\mathcal F(\mu*G_\varepsilon)=\hat\mu\,e^{-\varepsilon t^2}\). Also \(\mathcal F(g*G_\varepsilon)=(\mathcal Fg)\,e^{-\varepsilon t^2}\): this is Fubini's theorem when \(g\in L^1\cap L^2\), and both sides are continuous in \(g\in L^2\). As \(\mathcal F\) is injective, \(\mu*G_\varepsilon=g*G_\varepsilon\) almost everywhere. Let \(\phi\in C_c(\mathbb R)\). By Fubini's theorem and evenness, \(\int\phi\,(\mu*G_\varepsilon)\,d\lambda =\int(\phi*G_\varepsilon)\,d\mu\), which tends to \(\int\phi\,d\mu\), because \(\phi*G_\varepsilon\to\phi\) uniformly with \(|\phi*G_\varepsilon|\le\sup|\phi|\). And \(\int\phi\,(g*G_\varepsilon)\,d\lambda\to\int\phi g\,d\lambda\), because \(\|g*G_\varepsilon-g\|_2\le\int G_\varepsilon(s)\|g(\cdot-s)-g\|_2\,ds\to0\) by the continuity of translation in \(L^2\). So \(\int\phi\,d\mu=\int\phi g\,d\lambda\) for every \(\phi\in C_c(\mathbb R)\). For a bounded open interval \(I\), take \(\phi_n\in C_c(\mathbb R)\) with \(0\le\phi_n\uparrow1_I\); dominated convergence gives \(\mu(I)=\int_Ig\,d\lambda\). On a bounded interval \(J\), the Borel sets \(E\subseteq J\) with \(\mu(E)=\int_Eg\,d\lambda\) form a Dynkin class that contains the open subintervals, a family closed under finite intersections; so \(\mu(E)=\int_Eg\,d\lambda\) for every bounded Borel set \(E\), and then for every Borel set. Hence \(\mu=g\,d\lambda\), and \(\int|\hat\mu|^2dt=\int|\mathcal Fg|^2dt=2\pi\int|g|^2d\lambda\). \(\square\)

Proposition 8.3. Let \(Y,\mu,(A_y),P,\alpha\) be as in Proposition 7.1. Then \(\alpha\) is integrable iff the spectral measure of \(A_y\) is absolutely continuous (with respect to Lebesgue measure) for \(\mu\)-almost every \(y\).

Proof. One fibre. Let \(A\) act on \(K\) and \(a,\zeta\in K\). Then \(\langle e^{itA}aa^*e^{-itA}\zeta,\zeta\rangle= |\langle e^{itA}a,\zeta\rangle|^2=|\hat\mu_{a,\zeta}(t)|^2\), so \[ \langle E^\alpha(aa^*)\zeta,\zeta\rangle=\int|\hat\mu_{a,\zeta}(t)|^2\,dt. \tag{8.1} \] If \(\mu_a=w_a\,d\lambda\) with \(w_a\le N\), then \(\mu_{a,\zeta}\ll d\lambda\) and its density \(g\) satisfies \(|g|^2\le w_aw_\zeta\) almost everywhere (differentiate the inequality \(|\mu_{a,\zeta}(I)|^2\le\mu_a(I)\mu_\zeta(I)\) on intervals shrinking to a point; here \(\mu_{a,\zeta}=\mu_{a,P\zeta}\) with \(P\) the projection on the absolutely continuous subspace, so only the density of \(\mu_{P\zeta}\) enters). Hence \(\int|g|^2\le N\|\zeta\|^2\) and \(E^\alpha(aa^*)\le2\pi N\). If instead \(a\) has a nonzero component \(a_s\) in the singular subspace, take \(\zeta=a_s\): the singular subspace reduces \(A\), so \(\mu_{a,\zeta}=\mu_{a_s}\) is a nonzero singular measure, its Fourier transform is not square integrable, and (8.1) is infinite. Consequently, if \(b\in\mathcal L(K)_+\) is integrable, writing \(b=\sum_ja_ja_j^*\) we get \(E^\alpha(a_ja_j^*)\le E^\alpha(b)\) bounded, so every \(a_j\) is absolutely continuous and \(s(b)\) is below the projection on the absolutely continuous subspace.

Only if. By Lemma 8.1(2) and separability there are integrable \(B^n\in P_+\) with \(\bigvee s(B^n)=1\). Since \(\langle E^\alpha(B)\zeta,\zeta\rangle=\int\!\!\int\langle\alpha^y_t(B_y)\zeta_y,\zeta_y\rangle\,dt\,d\mu(y)\) by Tonelli, a bounded \(E^\alpha(B^n)\) forces \(E^{\alpha^y}(B^n_y)\) to be bounded for almost every \(y\) (test on a countable family of sections dense in every fibre). By the one-fibre result, for almost every \(y\) every \(s(B^n_y)\) lies below the absolutely continuous projection of \(A_y\); and \(\bigvee_ns(B^n_y)=1\) almost everywhere, so \(A_y\) is absolutely continuous.

If. Let \((\xi^m)\) be a countable family of Borel sections with \(\|\xi^m_y\|\le1\) for all \(y\), dense in the unit ball of every fibre (take the finite rational combinations of a fundamental sequence, each divided by \(\max(1,\|\cdot\|)\)), and \(\mu^m_y=\mu_{\xi^m_y}\). The function \(w^m(y,\lambda)=\limsup_k2^k\mu^m_y(D_k(\lambda))\), with \(D_k(\lambda)\) the dyadic interval of length \(2^{-k}\) containing \(\lambda\), is Borel in \((y,\lambda)\) and is a density of \(\mu^m_y\) when \(\mu^m_y\) is absolutely continuous (differentiation theorem). Put \(a^{m,N}_y=1_{\{w^m(y,\cdot)\le N\}}(A_y)\xi^m_y\), a Borel section of norm at most \(1\), and \(B^{m,N}=a^{m,N}a^{m,N*}\in P_+\). For almost every \(y\), \(\mu_{a^{m,N}_y}\) has density \(\le N\), so \(E^\alpha(B^{m,N})\le2\pi N\) by the one-fibre estimate and Tonelli. As \(N\to\infty\), \(a^{m,N}_y\to 1_{\{w^m<\infty\}}(A_y)\xi^m_y=\xi^m_y\), because \(\{w^m=\infty\}\) is Lebesgue null, hence \(\mu^m_y\)-null. So the supports of the \(B^{m,N}\) have join \(1\), and \(\alpha\) is integrable by Lemma 8.1. \(\square\)

Theorem 8.4. Let \(T\) be a nonsingular positive random operator of degree \(1\) and \(\varphi=\varphi_T\).

(a) If \(\sigma^\varphi\) is integrable, then the spectral measure of \(T_x\) is absolutely continuous for \(\Lambda\)-almost every \(x\).

(b) If \(T\) is diagonal (Example 3.3(b): \(H=L^2\circ F\) and \(T=M(\rho)\), \(\rho>0\)), the converse holds: \(\sigma^\varphi\) is integrable iff the spectral measure of \(T_x\) is absolutely continuous for \(\Lambda\)-almost every \(x\).

Proof. Fix a faithful \(\nu\) and let \(\psi=\rho^\nu_T=\varphi\circ E_\nu\) (Theorem 6.3), an fns weight on \(P_\nu\) with \(\sigma^\psi_t=\operatorname{Ad}\nu(T)^{it}\). By Proposition 8.3 with \(A_x=\log T_x\), \(\sigma^\psi\) is integrable iff \(\log T_x\) (equivalently \(T_x\), as \(\log\) is a diffeomorphism of \((0,\infty)\) onto \(\mathbb R\)) is absolutely continuous for \(\Lambda_\nu\)-almost every \(x\). The set \(S\) of \(x\) where this fails is saturated, since \(\log T_y\) is unitarily equivalent to a translate of \(\log T_x\). If \(S\) is contained in a Borel \(\Lambda_\nu\)-null set \(Z\), then \(S\subset[Z]_\nu\): for \(y\in S\), \(s_*(\nu^y)\) is a nonzero measure carried by the orbit of \(y\), which lies in \(S\subset Z\). By (R3), \(S\) is then \(\Lambda\)-negligible. So "\(\Lambda_\nu\)-almost every" may be replaced by "\(\Lambda\)-almost every".

(a) follows from Lemma 8.2(a) applied to \(E_\nu\).

(b) Take \(f_1>0\) with \(\nu*f_1=1\) (R4) and \(f_k=\min(kf_1,1)\). Then \(f_k\uparrow1\), \(\nu*f_k\le k\), so \(x_k=M(f_k)\) increases to \(1\) and \(E_\nu(x_k)=M(\nu*f_k)\le k\) (Proposition 6.5). As multiplication operators, \(x_k\) commutes with \(\nu(T)=M(\rho)\), hence lies in the centralizer of \(\psi\). Lemma 8.2(b) shows that \(\sigma^\varphi\) is integrable when \(\sigma^\psi\) is. \(\square\)

Example 8.5. On \(L^\nu\) take \(T=M(\delta^{-1})\) (Example 3.3(c)). The spectral measure of \(T_x\) is equivalent to the image of \(\nu^x\) under \(\gamma\mapsto\delta(\gamma)^{-1}\). If \(\delta=1\) this is a point mass, and \(\sigma^{\varphi_T}\) is trivial and not integrable (unless \(H=0\)). If the image measures are absolutely continuous, \(\sigma^{\varphi_T}\) is integrable by Theorem 8.4(b). This happens, for instance, for the groupoid \(X\rtimes\mathbb R\) of a flow, with \(\delta(x,t)=e^t\) and \(\nu^x\) Lebesgue measure in the parameter \(t\): the image of Lebesgue measure under \(t\mapsto e^{-t}\) is equivalent to Lebesgue measure on \((0,\infty)\).

Now suppose that \(\sigma^\varphi\) is integrable, \(\varphi=\varphi_T\) faithful. Discarding a negligible saturated set, \(A_x=\log T_x\) is absolutely continuous for every \(x\). By (B6) we may write \(H_x=\int^\oplus_{\mathbb R} H'_{(x,s)}\,ds\), with \(A_x\) acting as multiplication by \(s\), measurably in \(x\). For \(\gamma:x\to y\), Lemma 3.2 gives \(U(\gamma)g(A_x)U(\gamma)^*=g(A_y+c(\gamma))\) for bounded Borel \(g\) (\(c=\log\delta\)): \(U(\gamma)\) carries multiplication by \(g(s)\) to multiplication by \(g(s+c(\gamma))\). Composing with the shift by \(c(\gamma)\) we obtain an operator commuting with the diagonal algebra, hence decomposable (B5): there are unitaries \[ U'(\gamma,s):H'_{(x,s+c(\gamma))}\to H'_{(y,s)},\qquad (U(\gamma)\zeta)(s)=U'(\gamma,s)\,\zeta(s+c(\gamma)), \] for almost every \(s\). A Borel choice in \((\gamma,s)\) exists: apply the last clause of (B6) over \(Y=G\) (with the measure \(m_\nu\), \(\nu\) faithful) to the field \(\gamma\mapsto U(\gamma)\), which intertwines \(\log T_{s(\gamma)}- c(\gamma)\) with \(\log T_{r(\gamma)}\). In the stable kernel \(G'\) of (R6), \((\gamma,s)\) is an arrow from \((x,s+c(\gamma))\) to \((y,s)\), so \(U'\) is a Borel family of unitaries along the arrows of \(G'\); since \(U\) is multiplicative, for each composable pair \((\gamma_1,\gamma_2)\) the relation \(U'(\gamma_1\gamma_2,s)=U'(\gamma_1,s) U'(\gamma_2,s+c(\gamma_1))\) holds for almost every \(s\), hence (Fubini) for almost every composable pair of \(G'\). Let \(\mu=\Lambda_\nu\times ds=\Lambda'_{\nu'}\) and \(m'=\mu\circ\nu'\). Since \(U'(\gamma')\) is a unitary from \(H'_{s(\gamma')}\) onto \(H'_{r(\gamma')}\) for \(m'\)-almost every arrow \(\gamma'\), the Borel function \(d=\dim H'_{(\cdot)}\) with values in \(\{0,1,\dots,\infty\}\) satisfies \(d\circ r=d\circ s\) \(m'\)-almost everywhere. This does not make the sets \(\{d=n\}\) saturated after removal of a negligible set, because a field can be changed on a null set that meets every orbit. Put instead \[ Y_n=\{y\in G'^{(0)}:\nu'^y(s^{-1}(\{d\neq n\}))=0\}. \] These sets are Borel, by the measurability of the kernel \(\nu'\) (R1). They are saturated: for an arrow \(\gamma'_0:y\to z\), left invariance gives \(\nu'^z=\gamma'_0\nu'^y\), and \(s(\gamma'_0\gamma')=s(\gamma')\). They are disjoint, because \(\nu'^y\neq0\) for every \(y\). Since \(d(s(\gamma'))=d(y)\) for \(\nu'^y\)-almost every \(\gamma'\), for \(\mu\)-almost every \(y\), almost every \(y\) lies in \(Y_{d(y)}\). So the complement of \(\bigcup_nY_n\) is a saturated \(\mu\)-null Borel set, \(\Lambda'\)-negligible by (R3), and \(d=n\) almost everywhere on \(Y_n\). On \(Y_n\) replace \(H'\) by the constant field \(\ell^2_n\) (\(\mathbb C^n\), or \(\ell^2(\mathbb N)\) for \(n=\infty\)), identified with \(H'\) on the Borel set \(Y_n\cap\{d=n\}\) by a measurable field of unitaries \(W\) (Measurable fields of Hilbert spaces and their direct integrals, Theorem 5.1). Replace \(U'(\gamma')\) by \(W_{r(\gamma')}U'(\gamma')W_{s(\gamma')}^{-1}\) when both ends of \(\gamma'\) lie in \(Y_n\cap\{d=n\}\), and by \(1\) otherwise. The arrows with an end in the null set \(Y_n\setminus\{d=n\}\) form an \(m'\)-null set, because \(m'\) and its image under \(\gamma'\mapsto\gamma'^{-1}\) have the same null sets (the modular relation (1.1) for \(\Lambda'\)). So the new family is still multiplicative for almost every composable pair, and on the reduction of \(G'\) to each \(Y_n\) it is a Borel map into the unitary group of \(\ell^2_n\), a Polish group. For \(\Lambda_\nu\)-almost every \(x\) only a null set of \(s\) is affected, so the decomposition \(H_x=\int^\oplus H'_{(x,s)}\,ds\) is unchanged. We keep the notation \(H'\) and \(U'\). The measure \(\mu\) and the faithful lift \(\nu'\) satisfy the hypothesis of (B7), by the modular relation (1.1) for \(\Lambda'\). So (B7), applied to the reduction of \(G'\) to each \(Y_n\), yields a genuine representation, still denoted \(U'\), of \(G'\) on \(H'\), equal to the family above almost everywhere, off a \(\mu\)-null saturated set, which is \(\Lambda'\)-negligible by (R3). For a section \(\xi\) of \(H\) we write \(\xi'_{(x,s)}=\xi_x(s)\).

Lemma 8.6. With these notations, let \(\nu\in\mathcal E^+\), \(\nu'\) its lift to \(G'\) (R6), and \(\xi\in D(U,\nu)\). For almost every \(y\) and every \(\alpha\in H_y=\int^\oplus H'_{(y,s)}ds\), \[ \big\langle E^\varphi(\theta_\nu(\xi,\xi))_y\alpha,\alpha\big\rangle=2\pi\int ds\int\big|\langle\alpha(s),U'(\gamma,s) \xi'_{s(\gamma,s)}\rangle\big|^2\,d\nu'^{(y,s)}(\gamma,s). \tag{8.2} \] In particular, if \(E^\varphi(\theta_\nu(\xi,\xi))\le c\), then for almost every \((y,s)\) (for \(\Lambda'_{\nu'}\)), \(\int|\langle\beta,U'(\gamma')\xi'_{s(\gamma')}\rangle|^2\,d\nu'^{(y,s)}(\gamma')\le(c/2\pi)\|\beta\|^2\) for all \(\beta\in H'_{(y,s)}\). If moreover \(\nu\) is faithful, then after removal of a \(\Lambda'\)-negligible saturated set, \(g\xi'\in D(U',\nu')\) for every Borel \(g\) on \(G'^{(0)}\) with \(0\le g\le1\) and \(g\xi'\) bounded.

Proof. By Corollary 5.4(a), \((\sigma^\varphi_t(X))_y=T_y^{it}X_yT_y^{-it}\), and by (1.2), \[ \langle E^\varphi(\theta_\nu(\xi,\xi))_y\alpha,\alpha\rangle=\int dt\int|\langle T_y^{-it}\alpha,U(\gamma) \xi_{s(\gamma)}\rangle|^2d\nu^y(\gamma)=\int d\nu^y(\gamma)\int|\langle T_y^{-it}\alpha,\beta_\gamma\rangle|^2dt, \] with \(\beta_\gamma=U(\gamma)\xi_{s(\gamma)}\). In the decomposition of \(H_y\), \(T_y^{-it}\alpha=(e^{-its}\alpha(s))\), so \(\langle T_y^{-it}\alpha,\beta_\gamma\rangle=\int e^{-its}\langle\alpha(s),\beta_\gamma(s)\rangle ds\) is the Fourier transform of the integrable function \(s\mapsto\langle\alpha(s),\beta_\gamma(s)\rangle\). By Plancherel (valid in \([0,\infty]\) for integrable functions), \(\int|\cdot|^2dt=2\pi\int|\langle\alpha(s),\beta_\gamma(s)\rangle|^2ds\), and \(\beta_\gamma(s)=U'(\gamma,s)\xi'_{(x,s+c(\gamma))}=U'(\gamma,s)\xi'_{s(\gamma,s)}\). Exchanging the integrals and using that \(\nu'^{(y,s)}\) is the image of \(\nu^y\) under \(\gamma\mapsto(\gamma,s)\) gives (8.2). If the left side is at most \(c\|\alpha\|^2\), the decomposable quadratic form on the right is bounded by \(c\), so its fibres are bounded by \(c/2\pi\) almost everywhere (test on a countable dense family of sections). The set of points of \(G'^{(0)}\) where this bound holds is saturated: for \(\gamma'_0:(y_0,s_0)\to(y,s)\), left invariance of \(\nu'\) and unitarity of \(U'(\gamma'_0)\) turn the integral at \((y,s)\) with the vector \(\beta\) into the integral at \((y_0,s_0)\) with \(U'(\gamma'_0)^*\beta\). So its complement is a saturated \(\Lambda'_{\nu'}\)-null set, \(\Lambda'\)-negligible by (R3) when \(\nu'\) is faithful; off it the bound holds everywhere, which is the condition defining \(D(U',\nu')\). Finally, multiplying by \(g\) only decreases the integrals. \(\square\)

The constant \(2\pi\) comes from the normalizations \(E^\varphi=\int\sigma_t\,dt\) and \(ds\) on \(\mathbb R=\log \mathbb R_+^*\).

Theorem 8.7. Let \(\varphi=\varphi_T\) be faithful with \(\sigma^\varphi\) integrable, and let \(H'\), \(U'\) be as above.

(a) After removal of a \(\Lambda'\)-negligible saturated set, \((H',U')\) is square integrable: it is a \(\Lambda'\)-random Hilbert space on the stable kernel \((G',\Lambda')\) of \(\delta\).

(b) The map \(A'\mapsto A\), \(A_x=\int^\oplus A'_{(x,s)}ds\), is an isomorphism of \(\operatorname{End}_{\Lambda'}(H')\) onto the centralizer \(\operatorname{End}_\Lambda(H)_\varphi\).

Proof. (a) By Lemma 8.1(3) there are \(y_n\uparrow1\) in \(\operatorname{End}_\Lambda(H)\) with \(E^\varphi(y_n)\) bounded; let \(A_n=y_n^{1/2}\), so \(A_n\to1\) strongly and \(E^\varphi(A_nA_n^*)\) is bounded. Let \(\nu\) be faithful and \(D\) a countable subset of \(D(U,\nu)\) with values total in every fibre (Q2). For \(\xi\in D\), \(A_n\xi\in D(U,\nu)\) and \(\theta_\nu(A_n\xi,A_n\xi)=A_n\theta_\nu(\xi,\xi)A_n^*\le\|\theta_\nu(\xi,\xi)\|A_nA_n^*\), so \(E^\varphi(\theta_\nu(A_n\xi,A_n\xi))\) is bounded. By Lemma 8.6, the truncations \(g_k(A_n\xi)'\), \(g_k= 1_{\{\|(A_n\xi)'\|\le k\}}\), lie in \(D(U',\nu')\). Since \(\nu(A_n)\to1\) strongly, a subsequence satisfies \(A_{n,x}\xi_x\to\xi_x\) for all \(\xi\in D\) and almost every \(x\); so for almost every \(x\) the vectors \(A_{n,x}\xi_x\) are total in \(H_x\), and then their components \((A_n\xi)'_{(x,s)}\) are total in \(H'_{(x,s)}\) for almost every \(s\) (a family of sections whose values span a proper subfield at a non-null set of \(s\) cannot be total in the direct integral). So the countable family of truncations has values total at \(\Lambda'_{\nu'}\)-almost every point of \(G'^{(0)}\). Let \(Z\) be the null set where this fails. For \(y\notin[Z]_{\nu'}\), \(s(\gamma')\notin Z\) for \(\nu'^y\)-almost every \(\gamma'\), and a single such \(\gamma'\) already makes \(\{U'(\gamma')\eta_{s(\gamma')}\}\) total. By the last clause of (Q2), applied on \(G'\) with the faithful \(\nu'\) and the countable family of truncations, the sections \(U'(f\nu')\eta\) form a countable subset of \(D(U',\nu')\) whose values are total at every point off \([Z]_{\nu'}\). By (R3), \([Z]_{\nu'}\) is \(\Lambda'\)-negligible, and \(U'\) is square integrable off it.

(b) Let \(A'\in\operatorname{End}_{\Lambda'}(H')\) (a bounded Borel intertwiner). Then \(A_x=\int^\oplus A'_{(x,s)}ds\) is bounded and Borel. The unitary \(U(\gamma)\) is the direct integral of the unitaries that decompose it, and these agree with the \(U'(\gamma,s)\) for almost every \(s\), for \(m_\nu\)-almost every \(\gamma\), since the genuine representation \(U'\) agrees with the family constructed above almost everywhere. So \(U(\gamma)A_x=A_yU(\gamma)\) for \(m_\nu\)-almost every \(\gamma\). By (Q5), as in the proof of surjectivity below, this almost-everywhere intertwiner defines an element of \(\operatorname{End}_\Lambda(H)\), equal to \((A_x)\) almost everywhere, which we still denote \(A\); \(A_x\) commutes with multiplication by bounded functions of \(s\), in particular with \(T_x^{it}\), which is multiplication by \(e^{its}\); so \(\sigma^\varphi_t(A)=A\) by Corollary 5.4(a). The map is a normal \(*\)-homomorphism. It is injective: if \(A=0\) as a random operator, then \(A'_{(x,s)}=0\) for \(\Lambda_\nu\times ds\)-almost every \((x,s)\), and the set where \(A'\neq0\) is saturated in \(G'^{(0)}\), hence \(\Lambda'\)-negligible (R3). It is surjective: if \(A\in\operatorname{End}_\Lambda(H)_\varphi\), then \(T_x^{it}A_xT_x^{-it}=A_x\) for all rational \(t\) and, by Lemma 3.4 and continuity, for all \(t\) and almost every \(x\); so \(A_x\) commutes with the unitary group of \(\log T_x\), hence with its spectral measure, and is decomposable, \(A_x=\int^\oplus A'_{(x,s)}ds\) (B6). The family \(A'\) is an intertwiner for \(U'\) almost everywhere, because \(A\) is one for \(U\), and it is a genuine element of \(\operatorname{End}_{\Lambda'}(H')\) by (Q5) applied on \(G'\) (the equivalence with \(W(\nu')\)-modules turns an almost-everywhere intertwiner, which gives a \(W(\nu')\)-linear operator, into a random operator). \(\square\)

Theorem 8.7 identifies the centralizer of an integrable weight on \(\operatorname{End}_\Lambda(H)\) with an algebra of random operators on the stable kernel of \(\delta\), whose transverse measure \(\Lambda'\) has modulus \(\delta\circ h\). This is the groupoid form of the continuous decomposition of [Connes–Takesaki 1977]: the centralizer of an integrable weight is semifinite, and its centre carries the flow of weights.

9. Integration formulas

In this section \(F\) is a proper random variable, \(X=\bigsqcup F(x)\), \(H=L^2\circ F\) with \((U(\gamma)g)(z)= \delta(\gamma)^{1/2}g(\gamma^{-1}z)\), and \(M(f)\) denotes multiplication by a Borel function \(f\ge0\) on \(X\). Every random Hilbert space is isomorphic to a subspace of such an \(H\) (Q2 and Example 3.3(c)), so the formulas below, combined with Proposition 5.5, compute \(\varphi_T\) in general.

Proposition 9.1. Let \(\nu_i\in\mathcal E^+\) and Borel \(f_i\ge0\) on \(X\) (\(i\) in a countable set) with \(\sum_i\nu_i*f_i=1\). For every positive random operator \(T\) of degree \(1\) and every \(A\in\operatorname{End} _\Lambda(H)_+\), \[ \varphi_T(A)=\sum_i\int\omega_{T_x}\big(A_x^{1/2}M(f_i)_xA_x^{1/2}\big)\,d\Lambda_{\nu_i}(x), \tag{9.1} \] where for unbounded \(f_i\) the integrand means \(\sup_k\omega_{T_x}(A_x^{1/2}M(\min(f_i,k))_xA_x^{1/2})\). Formally, \(\varphi_T(A)=\sum_i\int\operatorname{Trace}(A_x^{1/2}M(f_i)A_x^{1/2}T_x)\,d\Lambda_{\nu_i}(x)\).

Proof. Put \(f_{i,k}=\min(f_i,k)\). The family \(B_x=A_x^{1/2}M(f_{i,k})_xA_x^{1/2}\) is bounded, and since \(A\) is an intertwiner, its average (6.2) for \(\nu_i\) is \(A_y^{1/2}\big(\int U(\gamma)M(f_{i,k})U(\gamma)^*d\nu_i^y\big)A_y^{1/2}= A_y^{1/2}M(\nu_i*f_{i,k})_yA_y^{1/2}\) (by the computation in the proof of Proposition 6.5, which does not use faithfulness), and it is bounded by \(\|A\|\). By Proposition 6.2(d), \[ \varphi_T\big(A^{1/2}M(\nu_i*f_{i,k})A^{1/2}\big)=\int\omega_{T_x}\big(A_x^{1/2}M(f_{i,k})_xA_x^{1/2}\big)\, d\Lambda_{\nu_i}(x). \] As \(k\to\infty\), \(\sum_iA^{1/2}M(\nu_i*f_{i,k})A^{1/2}\) increases to \(A^{1/2}M(\sum_i\nu_i*f_i)A^{1/2}=A\). Normality of \(\varphi_T\) and monotone convergence give (9.1). \(\square\)

Lemma 9.2 (the local trace). For every positive random operator \(T\) of degree \(1\) on \(H=L^2\circ F\) there is a \(G\)-invariant Borel function \(w_T:X\to[0,\infty]\) such that, for every \(x\) and every Borel \(g\ge0\), \[ \alpha_T^x(g):=\sup_k\omega_{T_x}\big(M(\min(g,k))_x\big)=\int g\,w_T\,d\alpha^x . \] The measures \(\alpha^x_T\) satisfy \(\gamma_*\alpha^x_T=\delta(\gamma)\alpha^y_T\) for \(\gamma:x\to y\).

Proof. Let \((e^j)\) be a measurable orthonormal frame of \(H\) and \(T_{n,x}=T_x(1+T_x/n)^{-1}\). For bounded \(g\), \(\operatorname{Tr}(T_n^{1/2}M(g)T_n^{1/2})=\sum_j\langle M(g)T_n^{1/2}e^j,T_n^{1/2}e^j\rangle=\int g\,w_n\, d\alpha^x\) with \(w_n=\sum_j|T_n^{1/2}e^j|^2\), a Borel function on \(X\). These integrals increase with \(n\) for every \(g\ge0\), so \(w_n\le w_{n+1}\) \(\alpha^x\)-almost everywhere, and \(w=\sup_nw_n\) satisfies \(\omega_{T_x}(M(g))=\int gw\,d\alpha^x\) by monotone convergence; the case of unbounded \(g\) follows. For \(\gamma:x\to y\), \(U(\gamma)^*T_y U(\gamma)=\delta(\gamma)^{-1}T_x\) and \(U(\gamma)^*M(g)_yU(\gamma)=M(g\circ\gamma)_x\) give \(\alpha_T^y(g)= \delta(\gamma)^{-1}\alpha_T^x(g\circ\gamma)\), which is the transformation rule; the same rule for \(\alpha\) then shows \(w(\gamma^{-1}z)=w(z)\) for \(\alpha^y\)-almost every \(z\in F(y)\), for each \(\gamma\in G^y\). Now fix \(f_1>0\) with \(\nu*f_1=1\) (\(\nu\) faithful) and put \(w_T(z)=\int f_1(\gamma^{-1}z)w(\gamma^{-1}z)\, d\nu^{\pi(z)}(\gamma)\). It is \(G\)-invariant, like every function \(\nu*h\). By Tonelli on \(\nu^y\times\alpha^y\), for \(\alpha^y\)-almost every \(z\) we have \(w(\gamma^{-1}z)=w(z)\) for \(\nu^y\)-almost every \(\gamma\), hence \(w_T(z)=w(z)(\nu*f_1)(z)=w(z)\). So \(w_T\,\alpha^x=w\,\alpha^x=\alpha^x_T\). \(\square\)

We call \(F_T:x\mapsto(F(x),\alpha^x_T)\) the local trace of \(T\). The measures \(\alpha^x_T\) need not be \(\sigma\)-finite (for instance \(\alpha_T^x(g)=\operatorname{Trace}M(g)=\infty\) when \(T=1\) and \(\alpha^x\) has no atoms). But the truncations \(F_{T,n}:x\mapsto(F(x),\min(w_T,n)\alpha^x)\) are proper random variables of modulus \(\delta\) in the sense of (R4), increasing to \(F_T\), and we define \(\int F_T\,d\Lambda=\sup_n\int F_{T,n}\,d\Lambda\). By monotone convergence \(\int F_T\,d\Lambda=\int\alpha^x_T(f)\,d\Lambda_\nu(x)\) whenever \(\nu\) is faithful and \(\nu*f=1\).

Corollary 9.3. \(\varphi_T(1)=\int F_T\,d\Lambda\).

Proof. Take a faithful \(\nu\) and \(f\ge0\) with \(\nu*f=1\), and apply (9.1) with one term and \(A=1\): \(\varphi_T(1)=\int\alpha^x_T(f)\,d\Lambda_\nu(x)\). \(\square\)

So \(\varphi_T(1)=\int\operatorname{Trace}(T_x)\,d\Lambda(x)\) is the integral over the orbit space of the random variable "local trace of \(T\)", which is how the notation should be read.

Corollary 9.4 (convolution operators). Let \(\nu\in\mathcal E^+\) be faithful, \(E=(E_x)\) a measurable field of Hilbert spaces on \(G^{(0)}\), and \(H_x=L^2(G^x,\nu^x;s^*E)\), the square-integrable sections \(\xi\) with \(\xi(\gamma)\in E_{s(\gamma)}\), with \((U(\gamma)\xi)(\gamma')=\xi(\gamma^{-1}\gamma')\). Let \(\Delta^{-1}\) be the family of multiplications by \(\delta^{-1}\), a nonsingular positive random operator of degree \(1\). Let \(k\) be a Borel field with \(k(\eta)\in\mathcal L(E_{s(\eta)},E_{r(\eta)})\), and \(T\in\operatorname{End}_\Lambda(H)\) given by \[ (T_x\xi)(\gamma)=\int k(\gamma^{-1}\gamma')\xi(\gamma')\,d\nu^x(\gamma') \] for bounded sections \(\xi\) with \(\nu^x\)-finite support (the integral converging absolutely). Then \[ \varphi_{\Delta^{-1}}(T^*T)=\int_G\|k(\eta)\|_{HS}^2\,\delta(\eta)^{-1}\,dm_\nu(\eta),\qquad m_\nu=\Lambda_\nu\circ\nu . \] In the notation above, \(\int\operatorname{Trace}(\delta_x^{-1}|T_x|^2)\,d\Lambda(x)=\int\|k\|_{HS}^2\,\delta^{-1} \,d(\Lambda_\nu\circ\nu)\).

Proof. \(H\) is square integrable: with a measurable orthonormal frame of \(E\), \(H\) is a direct sum of regular representations \(L^{(1_{Y_j}\circ s)\nu}\), \(Y_j=\{\dim E\ge j\}\). The operator \(\Delta^{-1}\) has degree \(1\) by Example 3.3(c). Choose \(g>0\) with \(\nu(g)=1\) and put \(f=\tilde g\), \(f_k=\min(f,k)\). Multiplication by \(f\) on \(H_x\) averages to multiplication by \((\nu*f)(\gamma)=\int f(\gamma'^{-1}\gamma)d\nu^{r(\gamma)}(\gamma')= \nu(g)(s(\gamma))=1\) (substitute \(\gamma'=\gamma\eta\) and use left invariance). As in Proposition 9.1, with \(B_x=T_x^*M(f_k)T_x\) (average \(T^*M(\nu*f_k)T\)) and normality, \[ \varphi_{\Delta^{-1}}(T^*T)=\sup_k\int\omega_{\delta^{-1}}\big(T_x^*M(f_k)T_x\big)\,d\Lambda_\nu(x). \] With \(S_\varepsilon\) the multiplication by \(s_\varepsilon=\delta^{-1}(1+\varepsilon\delta^{-1})^{-1}\) and \(Y=M(f_k)^{1/2}T_x\), Lemma 6.1 gives \(\omega_{\delta^{-1}}(Y^*Y)=\sup_\varepsilon\|YS_\varepsilon^{1/2}\|_{HS}^2\). The operator \(YS_\varepsilon^{1/2}\) agrees on a dense subspace with the integral operator of kernel \(f_k(\gamma)^{1/2}k(\gamma^{-1}\gamma')s_\varepsilon(\gamma')^{1/2}\); a bounded operator of this kind has Hilbert–Schmidt norm equal to the \(L^2\) norm of its kernel, both sides possibly infinite (if it is Hilbert–Schmidt it has an \(L^2\) kernel, which must agree almost everywhere with the given one; if the kernel is \(L^2\), the associated Hilbert–Schmidt operator agrees with it on a dense subspace). Letting \(\varepsilon\to0\) and \(k\to\infty\) (monotone convergence), and substituting \(\gamma'=\gamma\eta\), \[ \varphi_{\Delta^{-1}}(T^*T)=\int d\Lambda_\nu(x)\int d\nu^x(\gamma)\,f(\gamma)\delta(\gamma)^{-1}h(s(\gamma)),\qquad h(z)=\int\|k(\eta)\|_{HS}^2\delta(\eta)^{-1}d\nu^z(\eta). \] The right side is \(\int\Phi\,\delta^{-1}dm_\nu\) with \(\Phi(\gamma)=f(\gamma)h(s(\gamma))\). By the modular relation (R2), \(\int\Phi\delta^{-1}dm_\nu=\int\Phi(\gamma^{-1})dm_\nu(\gamma)=\int h(y)\big(\int g\,d\nu^y\big)d\Lambda_\nu(y)= \int h\,d\Lambda_\nu\), which is the claimed integral. \(\square\)

Proposition 9.5 (invariance under proper homomorphisms). Let \(\Lambda\), \(\Lambda'\) be \(\sigma\)-finite transverse measures of moduli \(\delta\), \(\delta'\) on \(G\), \(G'\), and \(h:G\to G'\) a proper homomorphism with \(h(\Lambda)=\Lambda'\) and \(\delta=\delta'\circ h\). Let \(H'\) be a \(\Lambda'\)-random Hilbert space, \(H=h^*H'\), \(T'\) a positive random operator of degree \(1\) on \(H'\) and \(T=h^*T'=(T'_{h(x)})\). Then \(T\) is a positive random operator of degree \(1\) on \(H\), and \[ \int\operatorname{Trace}(T_x)\,d\Lambda(x)=\int\operatorname{Trace}(T'_{x'})\,d\Lambda'(x'),\quad\text{that is,}\quad \varphi_T(1)=\varphi_{T'}(1). \]

Proof. \(H\) is \(\Lambda\)-random by (Q2). For \(\gamma:x\to y\), \(U(\gamma)T_x=U'(h(\gamma))T'_{h(x)}= \delta'(h(\gamma))T'_{h(y)}U'(h(\gamma))=\delta(\gamma)T_yU(\gamma)\).

Reduction. By (Q2) and Example 3.3(c) there are a proper random variable \(F'\) on \(G'\) and an isometry \(V\in\operatorname{Hom}_{\Lambda'}(H',L^2\circ F')\) (take \(F'=\mathbb N\times(x\mapsto(G'^x,\delta'^{-1}\nu'^x))\)). By Proposition 5.5(a), \(\varphi_{VT'V^*}(1)=\varphi_{T'}(V^*V)=\varphi_{T'}(1)\). Likewise \(h^*V\) is an isometry from \(H\) into \(h^*(L^2\circ F')=L^2\circ h^*F'\), with \(h^*(VT'V^*)=(h^*V)T(h^*V)^*\), so the value at \(1\) is unchanged on the left too. So we may assume \(H'=L^2\circ F'\) and \(H=L^2\circ h^*F'\); \(h^*F'\) is proper (R5).

Computation. Since \(F(x)=F'(h(x))\) and \(T_x=T'_{h(x)}\), the local traces satisfy \(\alpha^x_T=\alpha'^{h(x)}_{T'}\), and with \(w_T=w_{T'}\) (on \(F(x)=F'(h(x))\)) the truncations satisfy \(F_{T,n}=h^*F'_{T',n}\). By (R5), \(\int F_{T,n}\,d\Lambda=\int F'_{T',n}\,d\Lambda'\) for every \(n\). Taking suprema and using Corollary 9.3 on both sides gives the result. Here \(w_{T'}\), read on \(F(x)=F'(h(x))\), is \(G\)-invariant because \(G\) acts on \(\bigsqcup_xF'(h(x))\) through \(h\); so it is an admissible choice of \(w_T\), and the truncations are random variables on \(G\). \(\square\)

10. Formal dimension

In this section \(\delta=1\). Then the identity family \(1\) is a positive random operator of degree \(1\) on every random Hilbert space.

Definition 10.1. The formal dimension of a \(\Lambda\)-random Hilbert space \(H\) is \[ \dim_\Lambda(H)=\varphi_1(1)=\int\operatorname{Trace}(1_{H_x})\,d\Lambda(x)\in[0,\infty]. \] The weight \(\operatorname{Tr}_\Lambda=\varphi_1\) on \(\operatorname{End}_\Lambda(H)\) is characterized by \(\operatorname{Tr}_\Lambda(\theta_\nu(\xi,\xi))=\int\|\xi_x\|^2\,d\Lambda_\nu(x)\).

Proposition 10.2. (a) \(\operatorname{Tr}_\Lambda\) is a faithful normal semifinite trace on \(\operatorname{End}_\Lambda(H)\), and \(\operatorname{Tr}_\Lambda(p)=\dim_\Lambda(pH)\) for every projection \(p\).

(b) If \(\operatorname{Hom}_\Lambda(H_1,H_2)\) contains an invertible element, then \(\dim_\Lambda H_1=\dim_\Lambda H_2\).

(c) \(\dim_\Lambda\big(\bigoplus_iH_i\big)=\sum_i\dim_\Lambda H_i\) for countable direct sums.

(d) If \(\Lambda=\sum_i\Lambda_i\) is a countable sum of transverse measures (of modulus \(1\)), then \(\dim_\Lambda(H)=\sum_i\dim_{\Lambda_i}(H)\).

(e) If \(h:G\to G'\) is a proper homomorphism, \(\delta'=1\) and \(h(\Lambda)=\Lambda'\), then \(\dim_\Lambda(h^*H)=\dim_{\Lambda'}(H)\) for every \(\Lambda'\)-random Hilbert space \(H\).

Proof. (a) \(\varphi_1\) is normal, semifinite and faithful (Theorem 5.3), with \(\sigma_t^{\varphi_1}= \operatorname{Ad}1^{it}=\mathrm{id}\) (Corollary 5.4), so it is a trace (B2). Write \(H=pH\oplus(1-p)H\); by Proposition 5.5(b) with \(T=1\oplus1\), \(\varphi_1(p)=\varphi^{pH}_1(1)+\varphi^{(1-p)H}_1(0)=\dim_\Lambda(pH)\).

(b) Let \(A\) be invertible, \(A_x=u_x|A_x|\) the polar decompositions. The \(u_x\) are unitary, measurable (\(u_x=A_x |A_x|^{-1}\)), and intertwine, by uniqueness of the polar decomposition of \(U_2(\gamma)A_xU_1(\gamma)^*=A_y\). By Proposition 5.5(a) with \(V=u\), \(\varphi^{H_2}_{u1u^*}(1)=\varphi^{H_1}_1(u^*u)\), and \(u1u^*=1\).

(c) Proposition 5.5(b) with \(T=\bigoplus1\) and \(X=1\).

(d) \(H\) is \(\Lambda_i\)-random, and \(\pi_i:\operatorname{End}_\Lambda(H)\to\operatorname{End}_{\Lambda_i}(H)\), which keeps the family and forgets more null sets, is a normal \(*\)-homomorphism. The normal weight \(\sum_i\varphi_1^{\Lambda_i}\circ\pi_i\) takes the value \(\sum_i\int\|\xi\|^2d\Lambda_{i,\nu}=\int\|\xi\|^2d\Lambda_\nu\) on \(\theta_\nu(\xi,\xi)\), so it equals \(\varphi^\Lambda_1\) by Corollary 4.3. Evaluate at \(1\).

(e) Proposition 9.5 with \(T'=1\). \(\square\)

Example 10.3. In Example 5.6 (one orbit with mass \(c\)), \(\dim_\Lambda(H)=c\dim K\). For the trivial groupoid \(G=G^{(0)}=Y\) (only units), a transverse measure is a measure \(\mu\) on \(Y\), a random Hilbert space is a measurable field, and \(\dim_\Lambda(H)=\int\dim H_y\,d\mu(y)\). The formal dimension interpolates between these: it is an integral of dimensions over the orbit space, normalized by \(\Lambda\), and it takes all values in \([0,\infty]\).

Theorem 10.4 (unimodular groups). Let \(\Gamma\) be a unimodular locally compact second countable group, viewed as a groupoid with one unit, with \(\delta=1\). Fix a Haar measure \(dg\); the transverse functions are the multiples of \(dg\) (B9)(a), and \(\Lambda(c\,dg)=c\) defines a transverse measure. The \(\Lambda\)-random Hilbert spaces are the representations \(\pi\) of \(\Gamma\) contained in a multiple of the regular representation, and \(\operatorname{End}_\Lambda(\pi)=\pi(\Gamma)'\).

(a) If \(\pi\) is irreducible and square integrable with formal degree \(d_\pi\) (relative to \(dg\)), then \(\dim_\Lambda(\pi)=d_\pi\).

(b) On \(L^2(\Gamma)\), \(\operatorname{Tr}_\Lambda\) is the Plancherel trace \(\tau'\) of \(R(\Gamma)\); on \(L^2(\Gamma)\otimes\ell^2\), it is \(\tau'\otimes\operatorname{Tr}\). Hence, if \(\pi\) is realized as \(p(L^2(\Gamma)\otimes\ell^2)\) with \(p\in R(\Gamma)\otimes\mathcal L(\ell^2)\), then \(\dim_\Lambda(\pi)= (\tau'\otimes\operatorname{Tr})(p)\), the Murray–von Neumann dimension of \(\pi\) as a module over the group von Neumann algebra with its Plancherel trace.

Proof. (a) Let \(\xi\in K_\pi\) be a unit vector. By (B9)(c), \(\int|\langle\alpha,\pi(g)\xi\rangle|^2dg= d_\pi^{-1}\|\alpha\|^2\), so \(\xi\in D(\pi,dg)\) and \(\langle\theta_{dg}(\xi,\xi)\alpha,\alpha\rangle= d_\pi^{-1}\|\alpha\|^2\), that is \(\theta_{dg}(\xi,\xi)=d_\pi^{-1}1\). Hence \(\dim_\Lambda(\pi)=d_\pi\varphi_1 (\theta_{dg}(\xi,\xi))=d_\pi\|\xi\|^2\Lambda(dg)=d_\pi\).

(b) For \(\xi\in L^2(\Gamma)\) put \(\xi^*(u)=\overline{\xi(u^{-1})}\). Then \((T_{dg}(\xi)\alpha)(g)=\langle\alpha, \lambda(g)\xi\rangle=\int\alpha(u)\overline{\xi(g^{-1}u)}du=(\alpha*\xi^*)(g)\), so \(\xi\in D(\lambda,dg)\) iff right convolution \(\rho(\xi^*)\) is bounded, and then \(\theta_{dg}(\xi,\xi)=\rho(\xi^*)^*\rho(\xi^*)\) and \(\tau'(\theta_{dg}(\xi,\xi))=\|\xi^*\|^2=\|\xi\|^2\) (unimodularity). Also \(\operatorname{Tr}_\Lambda(\theta_{dg}(\xi,\xi)) =\|\xi\|^2\Lambda(dg)=\|\xi\|^2\). Both are normal weights on \(\operatorname{End}_\Lambda(\lambda)=R(\Gamma)\), so they agree (Corollary 4.3). Proposition 5.5(b) gives \(\operatorname{Tr}_\Lambda=\tau'\otimes\operatorname{Tr}\) on the multiple, and Proposition 10.2(a) gives \(\dim_\Lambda(pH)=(\tau'\otimes\operatorname{Tr})(p)\). \(\square\)

For a discrete group with counting measure, \(\dim_\Lambda(\ell^2\Gamma)=\tau'(1)=1\), and for a finite group \(\dim_\Lambda(\pi)=\dim\pi/|\Gamma|\) (Exercise 12.3).

Proposition 10.5 (transversals). Let \(Z\subset G^{(0)}\) be a transversal, that is, a Borel set whose characteristic transverse function \(\nu_Z\), \(\nu_Z(f)(y)=\sum_{\gamma\in G^y,\,s(\gamma)\in Z}f(\gamma)\), is proper. Let \(H_x=\ell^2(G^x\cap s^{-1}(Z))\) with left translations. Then \(\dim_\Lambda(H)=\Lambda(\nu_Z)\).

Proof. \(H=L^2\circ F\) for the random variable \(F(x)=G^x\cap s^{-1}(Z)\) with counting measure (of modulus \(1\)), a sub-functor of \(L\), hence proper (R4). For \(T=1\), \(\omega_1(M(g))=\operatorname{Trace}M(g)=\sum_{z\in F(x)}g(z)\), so \(F_1=F\), a \(\sigma\)-finite random variable, and \(\dim_\Lambda(H)=\int F\,d\Lambda\) by Corollary 9.3. Now \(F\) is the functor \(x\mapsto(G^x,\nu_Z^x)\). Take \(\nu\) faithful, \(g>0\) with \(\nu(g)=1\), and \(f=\tilde g\) on \(G\), so that \(\nu*f=1\) as in the proof of Corollary 9.4. Then, by the modular relation (1.1) with \(\delta=1\), \[ \int F\,d\Lambda=\int\nu_Z^x(\tilde g)\,d\Lambda_\nu(x)=\int\nu(g)\,d\Lambda_{\nu_Z}=\Lambda_{\nu_Z}(1)=\Lambda(\nu_Z). \qquad\square \]

For a countable measured equivalence relation \(R\) on \((Y,\mu)\) with invariant \(\mu\), the unit space itself is a transversal, \(\Lambda(\nu_Y)=\mu(Y)\), and \(\dim_\Lambda\) of \(\ell^2\) of the classes is the total mass \(\mu(Y)\): the formal dimension of \(\ell^2(\text{orbit})\) is the "number of points" of the orbit space, measured by \(\Lambda\).

11. \(\Lambda\)-compact operators and the \(\Lambda\)-index

In this section \(\delta=1\). The Fredholm theory relative to a trace goes back to Breuer; we give complete proofs, and Remark 11.8 notes that they give it for every semifinite von Neumann algebra. It is the framework in which the index theorem for measured foliations is stated [Connes 1982].

Fix \(\Lambda\)-random Hilbert spaces \(H_1,H_2,H_3\) and let \(M=\operatorname{End}_\Lambda(H_1\oplus H_2\oplus H_3)\) with the fns trace \(\tau=\operatorname{Tr}_\Lambda\) (Proposition 10.2). Let \(p_j\) be the projection onto \(H_j\), so that \(\operatorname{Hom}_\Lambda(H_j,H_k)=p_kMp_j\). By Proposition 5.5(b), the restriction of \(\tau\) to \(p_jMp_j=\operatorname{End}_\Lambda(H_j)\) is \(\operatorname{Tr}_\Lambda\) of \(H_j\), and by Proposition 10.2(a) \(\tau(q)=\dim_\Lambda(qH)\) for projections \(q\). For \(x\in M\), \(l(x)\) and \(r(x)\) are its left and right supports. For \(x\in p_kMp_j\) the kernel projection is \(N(x)=p_j-r(x)\), the projection onto \(\ker x\cap H_j\); thus \(\dim_\Lambda\operatorname{Ker}x=\tau(N(x))\), and \(N(x^*)=p_k-l(x)\). For \(a>0\), \(e_a(|x|)\) is the spectral projection of \(|x|\) for \([a,\infty)\).

Lemma 11.1. (a) \(\mathcal F=\{x\in M:\tau(l(x))<\infty\}\) is a two-sided \(*\)-ideal, and \(\tau(l(x))= \tau(r(x))\).

(b) Let \(\mathcal K\) be the norm closure of \(\mathcal F\). If \(x\in\mathcal K\) and \(e\) is a projection with \(\|x\xi\|\ge c\|\xi\|\) for \(\xi\in eH\) (some \(c>0\)), then \(\tau(e)<\infty\).

(c) \(\mathcal K=\{x\in M:\tau(e_a(|x|))<\infty\text{ for all }a>0\}\), and \(\mathcal K\) is also the norm closure of the ideal of definition \(\mathfrak m_\tau\) of \(\tau\).

Proof. (a) \(l(x)\sim r(x)\) (B8) and \(\tau\) is a trace. Since \(l(x^*)=r(x)\), \(\mathcal F\) is self-adjoint. For \(a\in M\), \(l(xa)\le l(x)\) and \(\tau(l(ax))=\tau(r(ax))\le\tau(r(x))\). Finally \(l(x+y)\le l(x)\vee l(y)\) and \(\tau(p\vee q)\le\tau(p)+\tau(q)\), since \(p\vee q-q\sim p-p\wedge q\le p\) (B8).

(b) Choose \(y\in\mathcal F\) with \(\|x-y\|<c\). For \(0\neq\xi\in eH\), \(\|y\xi\|\ge(c-\|x-y\|)\|\xi\|>0\), so \(ye\) is injective on \(eH\) and \(r(ye)=e\). Then \(\tau(e)=\tau(l(ye))\le\tau(l(y))<\infty\).

(c) If \(x\in\mathcal K\), then \(\||x|\xi\|\ge a\|\xi\|\) on the range of \(e_a(|x|)\), and \(\|x\xi\|=\||x|\xi\|\); apply (b). Conversely, if all \(\tau(e_a(|x|))\) are finite, then \(x_a=xe_a(|x|)\) has \(r(x_a)\le e_a(|x|)\), so \(x_a\in \mathcal F\), and \(\|x-x_a\|=\|x(1-e_a(|x|))\|\le a\). For the last statement: \(\mathfrak m_\tau\) is a two-sided ideal containing the projections of finite trace, and \(y=l(y)y\) for \(y\in\mathcal F\), so \(\mathcal F\subset \mathfrak m_\tau\); conversely, for \(x\ge0\) with \(\tau(x)<\infty\), \(a\,e_a(x)\le x\) gives \(\tau(e_a(x))\le\tau(x)/a\), so \(\mathfrak m_\tau\subset\mathcal K\). \(\square\)

Definition 11.2. An operator \(x\in\operatorname{Hom}_\Lambda(H_j,H_k)\) is \(\Lambda\)-compact if \(x\in \mathcal K\), that is, \(\dim_\Lambda e_a(|x|)H_j<\infty\) for every \(a>0\). It is a \(\Lambda\)-Fredholm operator if there is \(y\in\operatorname{Hom}_\Lambda(H_k,H_j)\) (a parametrix) with \(yx-p_j\in\mathcal K\) and \(xy-p_k\in \mathcal K\). Its \(\Lambda\)-index is \[ \operatorname{Ind}_\Lambda(x)=\dim_\Lambda\operatorname{Ker}x-\dim_\Lambda\operatorname{Ker}x^*=\tau(N(x))-\tau(N(x^*)). \]

Lemma 11.3. Let \(x\in p_kMp_j\) be \(\Lambda\)-Fredholm with parametrix \(y\).

(a) \(\tau(N(x))\) and \(\tau(N(x^*))\) are finite, so \(\operatorname{Ind}_\Lambda(x)\in\mathbb R\) is well defined.

(b) There are a projection \(e\le p_j\) with \(\tau(p_j-e)<\infty\) and \(c>0\) such that \(\|x\xi\|\ge c\|\xi\|\) for \(\xi\in eH\).

Proof. Let \(k=p_j-yx\in\mathcal K\). (a) On \(N(x)H\), \(k\xi=\xi\); apply Lemma 11.1(b) with \(c=1\). The same argument for \(x^*\), whose parametrix is \(y^*\), gives \(\tau(N(x^*))<\infty\). (b) Choose \(z\in\mathcal F\) with \(\|k-z\|\le1/2\) and let \(e=p_j\wedge(1-r(z))\), the projection onto \(p_jH\cap\ker z\). For \(\xi\in eH\), \(\xi=yx\xi+(k-z)\xi\), so \(\|\xi\|\le\|y\|\|x\xi\|+\|\xi\|/2\) and \(\|x\xi\|\ge\|\xi\|/(2\|y\|+1)\). By (B8), \(p_j-e\sim p_j\vee(1-r(z))-(1-r(z))\le r(z)\), which has finite trace. \(\square\)

Proposition 11.4 (index formula). Let \(x\in p_kMp_j\) be \(\Lambda\)-Fredholm, \(e\le p_j\) a projection with \(\tau(p_j-e)<\infty\) such that \(x\) is bounded below on \(eH\), and \(f=l(xe)\), the projection onto the closed subspace \(x(eH)\). Then \(\tau(p_k-f)<\infty\) and \[ \operatorname{Ind}_\Lambda(x)=\tau(p_j-e)-\tau(p_k-f). \]

Proof. Let \(e'=p_j-e\), \(q'=p_k-f\), \(a=fxe\) and \(d=q'xe'\). Then \(q'xe=0\), and \(a\) maps \(eH\) bijectively onto \(fH\), with inverse \(a^{-1}\in eMf\) (B8). Finiteness: \(f\le l(x)\le f\vee l(xe')\), so \(q'=(p_k-l(x))+(l(x)-f)\) with \(\tau(p_k-l(x))=\tau(N(x^*))<\infty\) and \(l(x)-f\le f\vee l(xe')-f\sim l(xe')-f\wedge l(xe')\le l(xe')\sim r(xe')\le e'\).

Reduction to a diagonal operator. Let \(n=a^{-1}fxe'\), which maps \(e'H\) into \(eH\) and vanishes on \(eH\); hence \(n^2=0\) and \(g=p_j+n\) is invertible in \(p_jMp_j\) with inverse \(p_j-n\). With \(x_0=a+d\) we get \(x_0g=a+d+an+dn=fxe+q'xe'+fxe'=x\), because \(an=fxe'\) and \(dn=0\). Since \(\ker x=g^{-1}(\ker x_0)\), \(N(x)\sim N(x_0)\) (B8); since \(g^*\) is invertible, \(\ker x^*=\ker(g^*x_0^*)=\ker x_0^*\). Relative to \(p_j=e+e'\) and \(p_k=f+q'\), \(x_0=a\oplus d\) with \(a\) invertible, so \(N(x_0)=e'-r(d)\) and \(N(x_0^*)=q'-l(d)\). Therefore \[ \operatorname{Ind}_\Lambda(x)=\tau(e')-\tau(r(d))-\tau(q')+\tau(l(d))=\tau(e')-\tau(q'), \] using \(r(d)\sim l(d)\), both of finite trace. \(\square\)

Lemma 11.5. Let \(e\le p_j\) be a projection, \(x,w\in p_kMp_j\), and suppose \(\|x\xi\|\ge c\|\xi\|\) on \(eH\) and \(\|w\|<c/2\). Then \(\|(x+w)\xi\|\ge(c/2)\|\xi\|\) on \(eH\), and the projections \(f=l(xe)\), \(f_w=l((x+w)e)\) satisfy \(p_k-f\sim p_k-f_w\).

Proof. The lower bound is clear. For \(\xi\in eH\), the unit vector \(\eta=(x+w)\xi/\|(x+w)\xi\|\) is at distance at most \(\|w\xi\|/\|(x+w)\xi\|\le\|w\|/(c-\|w\|)<1\) from \(fH\), and \(x\xi/\|x\xi\|\) is at distance at most \(\|w\|/c<1\) from \(f_wH\). By (B8), \(\|f-f_w\|<1\), and \(p_k-f\), \(p_k-f_w\) are projections of \(p_kMp_k\) at distance \(<1\), hence equivalent. \(\square\)

Theorem 11.6. (a) If \(x_1\in\operatorname{Hom}_\Lambda(H_1,H_2)\) and \(x_2\in\operatorname{Hom}_\Lambda(H_2,H_3)\) are \(\Lambda\)-Fredholm, then so is \(x_2x_1\), and \(\operatorname{Ind}_\Lambda(x_2x_1)=\operatorname{Ind}_\Lambda(x_1) +\operatorname{Ind}_\Lambda(x_2)\).

(b) The \(\Lambda\)-Fredholm operators form a norm-open subset of \(\operatorname{Hom}_\Lambda(H_1,H_2)\), on which \(\operatorname{Ind}_\Lambda\) is locally constant (continuous for the discrete topology of \(\mathbb R\)).

(c) If \(x\) is \(\Lambda\)-Fredholm and \(k\) is \(\Lambda\)-compact, then \(x+k\) is \(\Lambda\)-Fredholm and \(\operatorname{Ind}_\Lambda(x+k)=\operatorname{Ind}_\Lambda(x)\).

Proof. (b) Let \(x\) have parametrix \(y\), \(k_1=p_1-yx\), \(k_2=p_2-xy\), and let \(e,c\) be as in Lemma 11.3(b). Let \(\|w\|<\min(c/2,1/\|y\|)\). Then \(g=p_1+yw\) and \(h=p_2+wy\) are invertible in \(p_1Mp_1\), \(p_2Mp_2\), and \(g^{-1}y(x+w)=p_1-g^{-1}k_1\), \((x+w)yh^{-1}=p_2-k_2h^{-1}\). So \(x+w\) has a left parametrix \(y_L\) and a right parametrix \(y_R\) modulo \(\mathcal K\); then \(y_L\equiv y_L(x+w)y_R\equiv y_R\) modulo \(\mathcal K\), and \(y_L\) is a parametrix. By Lemma 11.5, \(x+w\) is bounded below on \(eH\) and \(\tau(p_2-l((x+w)e))=\tau(p_2-l(xe))\); by Proposition 11.4 applied to \(x\) and to \(x+w\) with the same \(e\), the indices agree.

(c) \(y\) is also a parametrix of \(x+k\). Choose \(z\in\mathcal F\) with \(\|k-z\|<c/2\); replacing \(z\) by \(p_2zp_1\in\mathcal F\) we may assume \(z\in p_2Mp_1\). Let \(e_2=e\wedge(p_1-r(z))\). By (B8), \(e-e_2\sim e\vee(p_1-r(z))- (p_1-r(z))\le r(z)\), so \(\tau(p_1-e_2)<\infty\). On \(e_2H\), \(z\) vanishes, so \((x+k)\xi=(x+w)\xi\) with \(w=(k-z)e_2\), \(\|w\|<c/2\), and \(x\) is bounded below by \(c\) there. Lemma 11.5 and Proposition 11.4 (for \(x\) and for \(x+k\), with \(e_2\)) give equal indices.

(a) Parametrices \(y_1,y_2\) give \(y_1y_2x_2x_1=y_1(p_2-k)x_1\in p_1+\mathcal K\) and similarly on the other side, so \(x_2x_1\) is \(\Lambda\)-Fredholm. Choose \(e_1\le p_1\) for \(x_1\) and \(e_2\le p_2\) for \(x_2\) as in Lemma 11.3(b), and let \(f_1=l(x_1e_1)\), \(a_1=f_1x_1e_1\). Put \(g=f_1\wedge e_2\); then \(f_1-g\sim f_1\vee e_2-e_2\le p_2-e_2\), so \(\tau(f_1-g)<\infty\). Let \(e\le e_1\) be the projection onto \(a_1^{-1}(gH)\). Since \(e_1H\ominus a_1^{-1}(gH)= a_1^*(f_1H\ominus gH)\) and \(a_1^*\) is injective and bounded below on \(f_1H\), (B8) gives \(e_1-e\sim f_1-g\). So \(\tau(p_1-e)=\tau(p_1-e_1)+\tau(f_1-g)<\infty\). The operator \(x_1\) maps \(eH\) bijectively onto \(gH\), bounded below, and \(x_2\) is bounded below on \(gH\subset e_2H\); so \(x_2x_1\) is bounded below on \(eH\), with \(l(x_2x_1e)=l(x_2g)=:h\). Proposition 11.4 gives \[ \operatorname{Ind}(x_2x_1)=\tau(p_1-e_1)+\tau(f_1-g)-\tau(p_3-h),\quad\operatorname{Ind}(x_1)=\tau(p_1-e_1)-\tau(p_2-f_1), \quad\operatorname{Ind}(x_2)=\tau(p_2-g)-\tau(p_3-h), \] the last one using \(g\le e_2\) (so \(x_2\) is bounded below on \(gH\)) and \(\tau(p_2-g)=\tau(p_2-f_1)+\tau(f_1-g)< \infty\). Adding the last two gives the first. \(\square\)

Clearly \(\operatorname{Ind}_\Lambda(x^*)=-\operatorname{Ind}_\Lambda(x)\).

Remark 11.8 (Breuer's theory in general). The proofs of Lemma 11.1 to Theorem 11.6 use only that \(M\) is a von Neumann algebra with a faithful normal semifinite trace \(\tau\), that \(p_1,p_2,p_3\) are projections of \(M\) with sum \(1\), and the facts (B8), which hold in every von Neumann algebra. So they prove the same statements in that generality. For a von Neumann algebra \(N\) with a faithful normal semifinite trace \(\tau_N\), apply them to the algebra \(M_3(N)\) of \(3\times3\) matrices over \(N\), with the trace \(\tau_N\otimes\operatorname{Tr}\) and the diagonal matrix units \(p_j\): each corner \(p_kM_3(N)p_j\) is a copy of \(N\), and one obtains Breuer's Fredholm theory for operators in \(N\) relative to \(\tau_N\).

Example 11.7 (non-integral index). Let \(G=Y=[0,1]\) (only units) with Lebesgue measure as \(\Lambda\), \(H_y= \ell^2(\mathbb N)\) for all \(y\), and \(S\) the unilateral shift. Fix \(t\in[0,1]\) and let \(x_y=S^*\) for \(y\le t\), \(x_y=1\) for \(y>t\). A parametrix is \(y_y=S\) for \(y\le t\), \(1\) otherwise: \(yx-1\) and \(xy-1\) are the families \(-1_{[0,t]}(y)e_0e_0^*\) and \(0\), of finite trace. \(\operatorname{Ker}x_y=\mathbb Ce_0\) for \(y\le t\), and \(\operatorname{Ker}x_y^*=0\). So \(\operatorname{Ind}_\Lambda(x)=t\). In Example 5.6 (one orbit of mass \(c\)), \(\mathcal K\) is the ideal of compact operators on \(K\) and \(\operatorname{Ind}_\Lambda=c\cdot\operatorname{ind}\), the ordinary Fredholm index scaled by the mass of the orbit.

12. Exercises

Exercise 12.1. Let \(G=Y\) be a standard Borel space with only units, \(\Lambda=\mu\) a \(\sigma\)-finite measure, and \(H=(H_y)\) a measurable field. Show that \(\operatorname{End}_\Lambda(H)\) is the algebra of decomposable operators on \(\int^\oplus H_y\,d\mu\), that \(\varphi_T(A)=\int\omega_{T_y}(A_y)\,d\mu(y)\), and that \(\dim_\Lambda(H)=\int\dim H_y\,d\mu(y)\). Compute \(E_\nu\) for \(\nu^y=g(y)\varepsilon_y\), \(g>0\).

Solution. Here \(G^y=\{y\}\), every representation is square integrable, intertwiners are arbitrary bounded Borel families, and negligible sets are \(\mu\)-null sets; so \(\operatorname{End}_\Lambda(H)\) is the decomposable algebra. A transverse function is \(\nu^y=g(y)\varepsilon_y\) with \(g\ge0\) Borel, \(\Lambda_\nu=g\mu\), and \(\theta_\nu(\xi,\xi)_y=g(y)\xi_y\xi_y^*\). The weight \(A\mapsto\int\omega_{T_y}(A_y)d\mu\) is normal and takes the value \(\int g\langle T\xi,\xi\rangle d\mu=\int\langle T\xi,\xi\rangle d\Lambda_\nu\) on \(\theta_\nu(\xi,\xi)\), so it is \(\varphi_T\) by Corollary 4.3. With \(T=1\) and \(A=1\): \(\dim_\Lambda H=\int\operatorname{Tr}(1_{H_y})d\mu\). Here \(P_\nu=\operatorname{End}_\Lambda(H)\) and the average (6.2) of \(B\) is \(gB\); so \(E_\nu(B)=gB\), and \(\varphi_T(gB)= \int\omega_{T_y}(B_y)\,g\,d\mu=\rho^\nu_T(B)\), as Theorem 6.3 requires.

Exercise 12.2. In Example 5.6 (pair groupoid on \((X,\beta)\), \(\beta\) a probability measure, \(\Lambda(\nu_\beta)=c\)), identify every \(H_x\) with \(K\). Show that \(E_{\nu_\beta}\) is the conditional expectation \((B_x)\mapsto \int_XB_x\,d\beta(x)\) of \(P_{\nu_\beta}=L^\infty(X,\beta)\bar\otimes\mathcal L(K)\) onto \(\mathcal L(K)\), and that \(\rho^{\nu_\beta}_T=c\operatorname{Tr}(T_0^{1/2}E_{\nu_\beta}(\cdot)T_0^{1/2})\).

Solution. Under the identification, \(U(y,x)\) becomes the identity of \(K\), so the average (6.2) is \(C=\int B_x\,d\beta(x)\), constant in \(y\), bounded by \(\|B\|\) because \(\beta(X)=1\). By Theorem 6.3(b), \(E_{\nu_\beta}(B)=\int B_xd\beta\), and \(E(1)=1\). Then \(\rho_T(B)=\int\omega_{T_0}(B_x)\,c\,d\beta(x)= c\,\omega_{T_0}(\int B_xd\beta)\). Indeed, with \(S=T_0\) and \(S_\varepsilon\) as in Section 6, and an orthonormal basis \((e_i)\) of \(K\), \(\operatorname{Tr}(S_\varepsilon^{1/2}B_xS_\varepsilon^{1/2})=\sum_i\langle B_xS_\varepsilon^{1/2}e_i,S_\varepsilon^{1/2}e_i\rangle\); Tonelli's theorem for these nonnegative terms and the definition of the weak integral give \(\int\operatorname{Tr}(S_\varepsilon^{1/2}B_xS_\varepsilon^{1/2})\,d\beta(x)= \operatorname{Tr}(S_\varepsilon^{1/2}(\int B_xd\beta)S_\varepsilon^{1/2})\). Both sides increase as \(\varepsilon\downarrow0\), and monotone convergence along \(\varepsilon=1/n\) gives the claim. This is \(\varphi_T(E(B))\) by Example 5.6.

Exercise 12.3. Let \(\Gamma\) be a finite group with counting measure as Haar measure and \(\Lambda\) as in Theorem 10.4. Show that \(\dim_\Lambda(\pi)=\dim\pi/|\Gamma|\) for every irreducible \(\pi\), and check Proposition 10.2(c) on the regular representation.

Solution. Every representation is square integrable. Schur orthogonality for finite groups gives \(\sum_g|\langle\pi(g)\xi,\eta\rangle|^2=\frac{|\Gamma|}{\dim\pi}\|\xi\|^2\|\eta\|^2\), so \(d_\pi=\dim\pi/|\Gamma|\) and Theorem 10.4(a) gives \(\dim_\Lambda(\pi)=\dim\pi/|\Gamma|\). The regular representation is \(\bigoplus_\pi(\dim\pi)\,\pi\), so Proposition 10.2(c) gives \(\dim_\Lambda(\ell^2\Gamma)=\sum_\pi(\dim\pi)^2/|\Gamma|=1\), in agreement with \(\tau'(1)=\|\delta_e\|^2=1\) from Theorem 10.4(b).

Exercise 12.4. In Example 5.6 with \(\delta=1\), let \(T_0\) be a nonsingular positive operator on \(K\). Use Theorem 7.2 to show that \(\operatorname{Sp}(\sigma^{\varphi_T})\) is the closure of \(\operatorname{Sp}\log T_0- \operatorname{Sp}\log T_0\), and deduce the familiar fact that \(\operatorname{Tr}(T_0\,\cdot\,)\) is a trace iff \(T_0\) is scalar.

Solution. The isotropy groups of the pair groupoid are trivial, so Theorem 7.2 applies. \(T_x\) is unitarily equivalent to \(T_0\), so \(X_\varepsilon(p)\) is either empty or all of \(X\), and it is not negligible iff \((\operatorname{Sp}\log T_0- \operatorname{Sp}\log T_0)\cap(p-\varepsilon,p+\varepsilon)\neq\emptyset\). So \(p\in\operatorname{Sp}(\sigma^\varphi)\) iff \(p\) is in the closure of the difference set. By Corollary 7.3, \(\varphi_T=c\operatorname{Tr}(T_0^{1/2}\cdot T_0^{1/2})\) is a trace iff \(T_0\) is scalar.

Exercise 12.5. Let \(\delta=1\) and \(x\in\operatorname{End}_\Lambda(H)\) be \(\Lambda\)-Fredholm. Show that \(\operatorname{Ind}_\Lambda(x)=0\) when \(\operatorname{Tr}_\Lambda(1)<\infty\), whereas Example 11.7 gives every value in \([0,1]\) for \(\operatorname{Tr}_\Lambda(1)=\infty\).

Solution. Take \(H_1=H_2=H\), so \(p_1=p_2=1\), and let \(e\), \(f=l(xe)\) be as in Proposition 11.4 (they exist by Lemma 11.3(b)). If \(\tau(1)<\infty\), then \(\operatorname{Ind}_\Lambda(x)=\tau(1-e)-\tau(1-f)=\tau(f)-\tau(e)\), and \(e\sim f\) by (B8), because \(x\) is injective on \(eH\) with closed range \(fH\). So the index vanishes. In Example 11.7, \(H_y=\ell^2(\mathbb N)\) is infinite-dimensional, so \(\operatorname{Tr}_\Lambda(1)=\infty\), and the index is \(t\).

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