Six laboratories on expectations and operator-valued averages

This lesson develops six original problems from the mathematical phenomena behind the exercises at the end of Takesaki II, Chapter IX, §4. The problems are stated afresh and solved in full. Four points require corrections or qualifications: the null space of an expectation is a left ideal, simultaneous Hilbert-seminorm minimizers need not form a face, Euclidean modulation requires the Plancherel dual measure, and a group Fourier expansion must be interpreted in its proper \(L^1\) or \(L^2\) topology.

Throughout, inclusions of von Neumann algebras are unital. A projection of norm one onto a von Neumann subalgebra is therefore a conditional expectation by Full nonunital retraction theorem. Sums and integrals whose values can be unbounded are taken in the extended positive cone described in Infinite energies have bounded spectral tests.

The faithful corner of a normal expectation

Let \(N\subseteq M\) and let \(E:M\to N\) be a normal projection of norm one. Choose a faithful normal semifinite weight \(\psi\) on \(N\), whose existence is Every von Neumann algebra has such a weight, and put \(\Phi=\psi\circ E\). Normality is immediate. If \((f_i)\subseteq\mathfrak n_\psi\) is an increasing net of positive contractions converging strongly to \(1\), then

Thus \(f_i x f_i\) belongs to the finite definition algebra of \(\Phi\), and these compressions converge strongly to \(x\). Hence \(\Phi\) is semifinite. Because \(\psi\) is faithful, let \(\mathcal L_E\) denote the set of \(x\in M\) satisfying \(E(x^*x)=0\). Its support description is

where \(e=s(\Phi)\), by Normality produces a largest null projection through The faithful semifinite support corner. The handedness in (X4.2) matters: \(\mathcal L_E\) is stable under left multiplication, so its projection form is \(M(1-e)\).

For \(b\in N\), bimodularity gives

Consequently \(M(1-e)\) is stable under right multiplication by \(N\). Taking \(x=1-e\), first with \(b\) and then with \(b^*\), gives

Thus \(e\in M\cap N'\). Let \(z\) be the central support of \(e\) in \(N'\). In a faithful nondegenerate representation, \(\mathcal Z(N')=N'\cap N''=\mathcal Z(N)\), so \(1-z\in N\). Since \(1-z\leq1-e\), (X4.2) and the fact that \(E\) fixes \(N\) imply

The central support of \(e\) in \(N'\) is therefore \(1\). Also \(E(1-e)=0\), hence \(E(e)=1\).

Define

The algebra \(Ne\) has identity \(e\), and \(n\mapsto ne\) is faithful because the central support of \(e\) is \(1\). The map \(E_e\) is normal, positive, contractive and \(Ne\)-bimodular. If \(n\in N\), then \(E_e(ne)=E(ne)e=nE(e)e=ne\), so it is a projection onto \(Ne\). If \(x\in eMe\) and \(E_e(x^*x)=0\), central support gives \(E(x^*x)=0\); then \(x\in M(1-e)\), while \(x=xe\), and hence \(x=0\). Thus \(E_e\) is faithful.

We next record the three type consequences.

Semifiniteness. It is enough to work with the faithful corner (X4.6), because corners of semifinite algebras are semifinite and \(N\cong Ne\). Suppose that a nonzero central summand \(qN\) were of type III. The restriction of \(E\) to \(qMq\) is faithful. Let \(\tau\) be a faithful normal semifinite trace on \(qMq\), choose \(0\ne b\in(qMq)_+\) with \(\tau(b)<\infty\), and put \(a=b^{1/2}\). Faithfulness makes \(E(b)\ne0\), so for some \(\delta>0\) the spectral projection

For \(x\in fNf\), bimodularity gives \(xE(b)x^*=E(xbx^*)\), and (X4.7) yields

If a bounded net \(x_i\in fNf\) tends sigma-strongly to zero, the finite-trace estimate for \(a\in\mathfrak n_\tau\) makes \(a x_i^*\to0\) sigma-strongly. Normal complete positivity of \(E\), followed by (X4.8), gives \(x_i x_i^*\to0\) sigma-strongly. Hence involution is sigma-strongly continuous on the unit ball of \(fNf\). An infinite projection cannot have this property: an infinite matrix-unit row supplies a bounded net converging sigma-strongly to zero whose adjoints do not. Therefore \(f\) is finite, contradicting that the nonzero corner \(fNf\) lies in a type III algebra. There is no type III central summand, so \(N\) is semifinite.

Type I. Suppose now that \(M\) is type I. The preceding paragraph makes \(N\) semifinite. If \(N\) had a nonzero type II summand, a further nonzero sigma-finite finite corner would give a faithful normal tracial state \(\tau_N\) on a finite type II algebra \(N_0\). Restricting the faithful expectation gives

where \(M_0\) is still type I and \(\varphi\) is a faithful normal state. Choose a faithful normal semifinite trace \(\operatorname{Tr}_0\) on \(M_0\). There is an injective positive \(h\in L^1(M_0,\operatorname{Tr}_0)\) with \(\varphi(x)=\operatorname{Tr}_0(hx)\). For \(n\in N_0\), \(x\in M_0\), and \(y=E_0(x)\), bimodularity and traciality give

so \(N_0\subseteq\{h\}'\cap M_0\).

We use the elementary type-I density lemma: if \(A\) is type I and an injective positive \(h\) is integrable for a faithful semifinite trace, then \(\{h\}'\cap A\) is finite type I and every von Neumann subalgebra of it is type I. Here is the cutoff mechanism. In \(C=\{h\}'\cap A\), the projections

are central, increase strongly to \(1\), and satisfy \(\operatorname{Tr}_0(c_k)\leq k\operatorname{Tr}_0(h)<\infty\). Thus \(c_kAc_k\) is finite type I. Its homogeneous decomposition is a central sum of algebras \(L^\infty(X)\,\overline\otimes\,M_n\); the standard polynomial identity on each \(M_n\) shows that every von Neumann subalgebra of a homogeneous summand is type I. The central sum over \(n\), followed by \(c_k\uparrow1\), proves the lemma. Finiteness of \(C\) also follows directly because \(x\mapsto\operatorname{Tr}_0(hx)\) is a faithful finite trace on \(C\). Applied to (X4.10), the lemma contradicts the choice of \(N_0\). Hence \(N\) is type I.

Atomicity. Write RNP for the Radon--Nikodym property. Use the Banach-space characterization

Indeed, the predual of \(\bigoplus_i B(H_i)\) is the \(\ell^1\)-sum of the trace classes \(\mathcal S_1(H_i)\), which has the Radon--Nikodym property; conversely a diffuse summand gives the standard nonatomic \(L^1\) obstruction. The Radon--Nikodym property passes to closed subspaces. For a normal expectation \(F:A\to B\), the preadjoint embedding

is isometric, since restriction back to \(B\) recovers \(\omega\). If \(M\) is atomic, then so is the corner \(eMe\); (X4.12) makes \((Ne)_*\) a closed subspace of \((eMe)_*\). Equation (X4.11) makes \(Ne\), and therefore \(N\), atomic.

The formula sometimes printed as \((1-e)M\) cannot be correct. For the vector-state expectation from \(M_2(\mathbb C)\) to the scalars and the rank-one support \(p_\xi\), the null matrices are exactly those satisfying \(x\xi=0\), namely \(M_2(\mathbb C)(1-p_\xi)\), not \((1-p_\xi)M_2(\mathbb C)\).

Fixed points from invariant normal states

Let \(G\leq\operatorname{Aut}(M)\), and assume that its invariant normal states detect every nonzero square: for \(x\ne0\) there is a normal state \(\omega\) such that \(\omega\circ\alpha=\omega\) for every \(\alpha\in G\), and

Write \(N=M^G\). If \(\omega\) is invariant, naturality of supports gives \(\alpha(s(\omega))=s(\omega)\), so \(s(\omega)\in N\).

Choose a maximal family \(\mathcal F\) of invariant normal states with pairwise orthogonal supports, and define the arbitrary positive sum

Both sums are suprema of finite subsums. If \(1-p\ne0\), (X4.13) supplies an invariant state \(\theta\) with \(\theta(1-p)>0\). Compressing and normalizing \(\theta\) by the fixed projection \(1-p\) produces another invariant state whose support is orthogonal to every support in \(\mathcal F\), contradicting maximality. Thus \(p=1\), and \(\Psi\) is faithful.

For a finite \(\mathcal E\subseteq\mathcal F\), put \(p_{\mathcal E}=\sum_{\omega\in\mathcal E}s(\omega)\). Orthogonality gives

The finite compressions \(p_{\mathcal E}xp_{\mathcal E}\) prove semifiniteness of \(\Psi\) on \(M\). Since every \(p_{\mathcal E}\) belongs to \(N\), the same argument proves that \(\nu=\Psi|_{N_+}\) is semifinite.

The weight \(\Psi\) is \(G\)-invariant. For \(\alpha\in G\) and \(t\in\mathbb R\), functoriality of modular automorphisms yields

The modular expectation theorem The criterion and its exact setting now gives a faithful normal conditional expectation \(E:M\to N\) satisfying

For \(\alpha\in G\), the map \(E\alpha\) is another \(\Psi\)-preserving expectation onto \(N\); uniqueness in ME-01 gives \(E\alpha=E\).

It remains to justify uniqueness among all faithful normal invariant projections, rather than only among \(\Psi\)-preserving ones. Let \(P:M\to N\) be faithful, normal, of norm one, and satisfy \(P\alpha=P\). Set \(\Phi=\nu\circ P\). Then \(\Phi\) is a faithful normal semifinite \(G\)-invariant weight and \(\Phi|_N=\Psi|_N=\nu\). Both modular groups leave \(N\) invariant and restrict there to \(\sigma^\nu\). For

naturality under \(G\) gives \(\alpha(u_t)=u_t\), hence \(u_t\in N\). For every \(n\in N\), equality of the restricted modular groups gives

Thus \(u_t\in\mathcal Z(N)\). The center is fixed by \(\sigma^\nu\), and the cocycle is therefore a one-parameter group \(u_t=h^{it}\) for a positive injective operator affiliated with \(\mathcal Z(N)\). The fixed-density theorem The faithful invariant-weight equivalence gives \(\Phi=\Psi_h\). Restriction to \(N\) says \(\nu_h=\nu\), so uniqueness of the central density gives \(h=1\). Thus \(\Phi=\Psi\), \(P\) preserves \(\Psi\), and ME-01 gives \(P=E\).

We have proved the existence and uniqueness of the faithful normal norm-one projection onto \(M^G\) that is constant on every \(G\)-orbit.

Orbit hulls and simultaneous GNS minimizers

Keep the hypotheses of X4-02, but do not use the expectation constructed there. For \(x\in M\), write \(Gx=\{\alpha(x):\alpha\in G\}\), and let

This is a sigma-weakly compact convex subset of the norm ball of radius \(\lVert x\rVert\).

For an invariant normal state \(\omega\), let \((H_\omega,\pi_\omega,\Lambda_\omega)\) be its GNS representation. Use the seminorm \(\lVert y\rVert_\omega=\lVert\Lambda_\omega(y)\rVert\), so that

This seminorm is sigma-weakly lower semicontinuous: its closed balls are intersections of inequalities against the normal coefficient functionals \(y\mapsto\langle\Lambda_\omega(y),\eta\rangle\). Invariance gives a unitary representation

Let \(P_\omega\) be the projection onto its fixed-vector space. For a finite set \(F\) of invariant states, use the direct sum of the representations (X4.22). The closed convex hull of a unitary orbit has a unique least-norm vector, and that vector is the projection onto the fixed-vector space. Hence one net of finite convex combinations of \(\alpha(x)\) converges in every \(H_\omega\), \(\omega\in F\), to \(P_\omega\Lambda_\omega(x)\). Put \(m_\omega(x)=\min_{z\in K(x)}\lVert z\rVert_\omega\). Let \(K_F(x)\) be the set of \(y\in K(x)\) satisfying, for every \(\omega\in F\),

Thus \(K_F(x)\) is nonempty, compact and \(G\)-invariant. Lower semicontinuity makes it closed.

The assertion that \(K_F(x)\) must be a face is false. Take \(M=\mathbb C^2\), let \(G\) interchange the two coordinates, use the invariant average state, and set \(x=(1,-1)\). Its orbit hull is the segment from \(x\) to \(-x\), parameterized by \(-1\leq t\leq1\), whereas

The midpoint \(0=\tfrac12x+\tfrac12(-x)\) is not an endpoint, so its singleton is not a face of the segment. Face structure is unnecessary for the argument.

For the fixed \(x\), abbreviate \(K_F=K_F(x)\). These sets, ordered by finite \(F\), have the finite-intersection property because

Compactness gives a point in

For every \(y\in K_\infty(x)\) and every invariant normal state \(\omega\), uniqueness of the Hilbert-space least-norm vector gives

The same is true of \(\alpha(y)\). Therefore \(\omega((\alpha(y)-y)^*(\alpha(y)-y))=0\) for all invariant \(\omega\); (X4.13) gives \(\alpha(y)=y\). Thus \(y\in N\). If \(y,z\) both lie in \(K_\infty(x)\), (X4.27) applied to \(y-z\) and (X4.13) give \(y=z\). Conversely, if \(y\in K(x)\cap N\), then \(\Lambda_\omega(y)\) is a fixed vector in the closed convex hull of the orbit of \(\Lambda_\omega(x)\). Every vector in that hull has fixed component \(P_\omega\Lambda_\omega(x)\), so a fixed one must equal that component. Hence \(y\in K_\infty(x)\). Write the unique point as \(E(x)\). We have proved \(K(x)\cap N=K_\infty(x)\), and this set is the singleton

Equation (X4.27) proves linearity without choosing compatible averaging nets. Put \(a=E(x+y)\) and \(b=E(x)+E(y)\). For every invariant normal state \(\omega\), linearity of \(P_\omega\) gives

The separating hypothesis gives \(a=b\), so \(E\) is additive. Scalars are handled identically. The map fixes \(N\), and \(\lVert E(x)\rVert\leq\lVert x\rVert\) because \(E(x)\in K(x)\). Hence CE-004 makes \(E\) positive, completely positive and \(N\)-bimodular.

For every invariant normal state \(\omega\), the vector \(\Lambda_\omega(1)\) is fixed, so (X4.27) also gives

If \(0\leq x_i\uparrow x\), then \(0\leq E(x_i)\uparrow y\leq E(x)\). Normality of the invariant states and (X4.30) give \(\omega(E(x)-y)=0\) for all of them; (X4.13) gives \(y=E(x)\). Thus \(E\) is normal. If \(E(x^*x)=0\), (X4.30) and (X4.13) give \(x=0\), so \(E\) is faithful.

Finally, let \(P:M\to N\) be any normal norm-one projection satisfying \(P\alpha=P\). Normality implies

Since \(E(x)\in K(x)\) and \(P\) fixes \(N\), \(P(x)=P(E(x))=E(x)\). This proves uniqueness using only orbit geometry, normal-state separation and the norm-one projection theorem; modular theory was not used.

The Euclidean modulation average

Let \(H=L^2(\mathbb R,ds)\), let \(A=L^\infty(\mathbb R)\) act by multiplication, and put

Use the Fourier transform \(\widehat f(t)=\int_{\mathbb R}e^{-ist}f(s)\,ds\). The compatible dual Haar measure is \(d\widehat t=dt/(2\pi)\). Plancherel then reads

For \(x\in B(H)_+\), define a closed extended positive form by

It may take the value \(+\infty\). Translation of \(t\) leaves the form invariant under every \(u_r\), so its representing extended positive is affiliated with \(\{u_r:r\in\mathbb R\}'=A\). Denote it by

The hat records extended-positive integration, not a bounded weak operator integral. The representation is legitimate because (X4.34) is a nonnegative quadratic form and Fatou's lemma makes it lower semicontinuous in \(\eta\); a lower-semicontinuous nonnegative form is closed, with a possibly nondense finite domain exactly as allowed in \(\widehat A_+\).

Normality can be checked before taking the full integral. For a compact \(K\subseteq\mathbb R\), integration over \(K\) gives a normal positive functional of \(x\): its trace-class density is the Bochner integral of \(\theta_{u_t^*\eta,u_t^*\eta}\) against \(dt/(2\pi)\). This integrand is norm continuous in the trace class and \(K\) has finite measure. The full nonnegative continuous coefficient integral is the supremum of these compact integrals. Interchanging that supremum with an increasing-net supremum proves normality of every vector observation of \(T\). The same argument works with compact subsets of any LCA dual group, using the local Haar convention of The full multiplication algebra and local measure convention.

For a rank-one positive \(\theta_{\xi,\xi}\eta=\langle\eta,\xi\rangle\xi\), put \(c_{\xi,\eta}(t)=\langle u_t\xi,\eta\rangle\) and \(q_{\xi,\eta}=\langle T(\theta_{\xi,\xi})\eta,\eta\rangle\). Plancherel gives

The converse Plancherel test identifies the full domain here and in X4-05. On any LCA group, let \(h\in L^1\) and suppose \(\widehat h\in L^2\). Choose a positive compactly supported continuous approximate identity \((k_i)\), normalized by \(\int k_i=1\). Then \(h*k_i\in L^1\cap L^2\), \(|\widehat k_i|\leq1\), and Plancherel gives \(\|h*k_i\|_2=\|\widehat h\,\widehat k_i\|_2\leq\|\widehat h\|_2\). Since \(h*k_i\to h\) in \(L^1\), choose indices with summable \(L^1\) errors. The resulting sequence converges almost everywhere; Fatou gives \(h\in L^2\). The forward Plancherel identity now gives equality of the two norms. This argument uses a sequence selected for this one function, and requires no countable neighborhood basis.

For arbitrary \(\xi,\eta\in H\), the product \(h=\xi\overline\eta\) lies in \(L^1\) by Cauchy--Schwarz. Its Fourier coefficient is \(c_{\xi,\eta}\), up to character reflection. Thus (X4.36) holds on all of \(H\), with both sides infinite exactly when \(\xi\overline\eta\notin L^2\). It identifies the entire finite form domain, so

If \(x\geq0\) and \((e_j)\) is an orthonormal basis, put \(P_F=\sum_{j\in F}\theta_{e_j,e_j}\). The finite partial sums

combine (X4.37) with monotone convergence. For \(\varphi(M_f)=\int_{\mathbb R}f(s)\,ds\), the result is

Both sides may be infinite.

The form definition makes \(T\) additive, homogeneous, normal and \(A\)-bimodular. Equation (X4.39) makes it faithful. Rank-one operators formed from \(L^2\cap L^\infty\) have bounded output by (X4.37), and their finite-dimensional corners are sigma-weakly dense in \(B(H)\); hence \(T\) is semifinite. It is therefore a faithful normal semifinite operator-valued weight. Uniqueness from the scalar composite follows from Uniqueness from one scalar composite, including infinity.

If one integrates (X4.32) against ordinary \(dt\), (X4.37) acquires the factor \(2\pi\), so the composite becomes \(2\pi\operatorname{Tr}\). Thus the unqualified \(dt\) formula is correct only when the symbol \(dt\) has already been declared to mean the Plancherel dual Haar measure \(dt/(2\pi)\).

Modulation over a locally compact abelian group

Let \(G\) be a locally compact abelian group with Haar measure \(dg\). For \(\mathcal Ff(\chi)=\int_G\overline{\chi(s)}f(s)\,dg(s)\), choose the dual Haar measure \(d\chi\) on \(\widehat G\) so that

This is the normalization constructed in An onto Plancherel transform through Haar normalization and a finite example. On \(H=L^2(G,dg)\), let \(A=L^\infty(G)\) and

For \(x\in B(H)_+\), set

As in (X4.34), this means the extended closed form obtained by integrating the positive matrix coefficients. Fatou's lemma again proves lower semicontinuity and hence closedness. It is invariant under every \(u_\gamma\). Characters generate the full multiplication algebra, so \(\{u_\gamma:\gamma\in\widehat G\}'=A\); hence (X4.42) belongs to \(\widehat A_+\).

For arbitrary \(\xi,\eta\in H\), put \(c_{\xi,\eta}(\chi)=\langle u_\chi\xi,\eta\rangle\) and \(q_{\xi,\eta}=\langle T_G(\theta_{\xi,\xi})\eta,\eta\rangle\). Then

The converse Plancherel test just proved includes the infinite values and identifies the entire finite form domain. Therefore

For \(\varphi(M_f)=\int_G f\,dg\), finite-rank approximation as in (X4.38) now proves

Normality, faithfulness and \(A\)-bimodularity follow exactly as in X4-04. Vectors in \(C_c(G)\) are bounded and form a dense subspace of \(L^2(G)\); their finite-rank operators have bounded outputs, proving semifiniteness without a countability assumption. Thus (X4.42) is the unique faithful normal semifinite operator-valued weight with scalar composite \(\operatorname{Tr}\). Its normalization is intrinsic: rescaling \(dg\) by \(c\) rescales the Plancherel dual measure by \(c^{-1}\), leaving (X4.45) unchanged.

A discrete-group average and its Fourier coefficients

Let \(G\) be a countable discrete group, let \(H=\ell^2(G)\), and write \(\delta_s\) for the standard basis. Use

The two representations commute, and

Let \(\tau(y)=\langle y\delta_e,\delta_e\rangle\) be the canonical finite trace on \(M\).

For \(x\in B(H)_+\), define

the supremum of the finite positive partial sums in the extended positive cone. Reindexing makes this extended positive invariant under \(\operatorname{Ad}\rho\), so (X4.47) puts it in \(\widehat M_+\). Put \(v_r=\rho(r)^*\delta_e\). Then

The sum is normal, \(M\)-bimodular and faithful. For the matrix units \(e_{a,b}\delta_t=\mathbf1_{\{b\}}(t)\delta_a\), its finite-ideal extension gives

Thus finite matrix corners have bounded output and are sigma-weakly dense; \(T\) is semifinite. Equations (X4.48–49) therefore define the unique faithful normal semifinite operator-valued weight satisfying \(\widehat\tau\circ T=\operatorname{Tr}\).

There is a precise Fourier form, but it is not an unrestricted operator series for every positive \(x\in B(H)\). If \(x\in\mathcal S_1(H)_+\), then \(T(x)\) is a positive element of \(L^1(M,\tau)\). Extend linearly to \(\mathcal S_1(H)\), and define \(\widehat x(s)=\operatorname{Tr}(\lambda(s)^*x)\). For every \(s\in G\),

It suffices to check (X4.51) on \(e_{a,b}\), where both sides are \(\mathbf1_{\{s=ab^{-1}\}}\), and then use trace-norm continuity. Consequently the notation

means equality of all Fourier coefficients in \(L^1(M,\tau)\).

If in addition \(T(x)\in L^2(M,\tau)\), the unitaries \((\lambda(s))_{s\in G}\) form an orthonormal basis of that Hilbert space, and (X4.52) converges in \(L^2\):

For a finite linear combination of matrix units, the sum is finite by (X4.50). For a general positive bounded \(x\), the quantities \(\operatorname{Tr}(\lambda(s)^*x)\) need not exist and no such Fourier series is asserted.

This also detects the missing left side in the printed formula. When \(G\) is infinite and \(x=e_{e,e}\), (X4.50) gives

Thus replacing \(T(x)\) by \(x\) is false even on a rank-one positive.

The mathematical antecedents for X4-01–06 are Takesaki, Theory of Operator Algebras II, Chapter IX, Exercise 4(1)–(6), printed pages 233–235. The semifinite and type-I mechanisms in X4-01 also use the ideas of Takesaki, Theory of Operator Algebras I, Chapter V, Lemmas 2.27–2.29 and Theorem 2.30. The exposition, counterexamples, normalization checks, domain corrections and proofs in this lesson are independently written.