Uniform geometry of faithful weights

The Connes cocycle can compare weights by more than its real-time values. A contractive continuation through a lower strip defines an order, and allowing all strip widths produces an extended metric on faithful normal semifinite weights. We prove the order and metric arguments, identify the trace-density model, correct a factor of two in the printed model, and prove completeness.

The source section fixes a separable properly infinite von Neumann algebra. The arguments below use neither restriction: throughout, \(M\) is a von Neumann algebra and \(\mathcal W_0(M)\) is its set of faithful normal semifinite weights. The trace model separately assumes that \(M\) is semifinite. The Fourier-filter convention is the one fixed in OA-FLOW. The two spectral applications below have complete contour and Poisson arguments relative to explicit Fourier and analytic inputs; their prerequisites and the half-strip domination theorem are not proved here.

Contractive strips form an order

For \(\lambda>0\), put

For \(\varphi,\psi\in\mathcal W_0(M)\), write \(\varphi\preceq_\lambda\psi\) when

has a sigma-strongly continuous extension to \(S_\lambda\), holomorphic in its interior, such that

Local boundedness and scalar holomorphy make the interior extension norm-holomorphic by the norming-predual argument in CX-03. Consequently, products of two such extensions are holomorphic.

For each fixed \(\lambda\), the relation \(\preceq_\lambda\) is an order. Reflexivity follows from \([D\varphi:D\varphi]_t=1\). If \(\varphi_1\preceq_\lambda\varphi_2\) and \(\varphi_2\preceq_\lambda\varphi_3\), the ordered chain rule gives

The product of the two strip extensions is contractive, proving transitivity.

For antisymmetry, suppose both directions hold and denote the two lower-strip extensions by \(u_z\) and \(v_z\). On the real axis, \(v_t=u_t^*\). Let \(\omega\) be a normal state. The two scalar functions

have equal boundary values and glue across the real axis. The elementary Morera gluing argument from MA-08 makes the result holomorphic on \(-\lambda<\operatorname{Im}z<\lambda\). Its modulus is at most one, while its value at zero is one. The maximum-modulus principle therefore makes it constant. Hence \(\omega(v_t)=1\) for every normal state and every real \(t\). Normal states separate \(M\), so \(v_t=1\). Fixed-reference cocycle injectivity in CX-10 gives \(\varphi_1=\varphi_2\).

Write

The extensions for different widths agree on overlaps by scalar analytic uniqueness. Thus (UW.6) is equivalently a contractive holomorphic extension of (UW.2) to the closed lower half-plane.

The half-strip recovers pointwise order

We use the following exact form of the weight-domination theorem, recorded as OA-MOD-UW-DEP-DOMINATION. If \(\alpha,\beta\in\mathcal W_0(M)\) and \(C>0\), then

is equivalent to continuation of \([D\alpha:D\beta]_t\) through the strip \(-1/2\leq\operatorname{Im}z\leq0\), with lower-boundary norm at most \(C^{1/2}\). The real boundary consists of unitaries, so the bounded-strip maximum principle gives the corresponding bound throughout the strip. This is Takesaki II, VIII.3, Theorem 3.17; its complete analytic proof remains an explicit prerequisite rather than being hidden in the present argument.

Taking \(C=1\) gives the precise bridge

All values in (UW.7)--(UW.8) are extended nonnegative values. No subtraction of infinite weights is involved.

Infinite strips and the balanced spectral subspace

Let \(\Phi=\varphi_1\oplus\varphi_2\) be the balanced weight on \(N=M\mathbin{\overline\otimes}M_2\). The balanced-matrix formula SI-14 gives

We use the OA-FLOW Fourier-filter definition: if \(f=\mathcal F b\), then

The operator integral is weak-star. The spectrum is the hull of the ideal of Fourier functions whose filters annihilate \(x\). For the orbit \(\gamma_t(x)=e^{i\nu t}x\), with \(x\ne0\), (UW.50) gives \(\gamma_f(x)=f(\nu)x\), so its spectrum is \(\{\nu\}\). Smooth compactly supported Fourier functions separate any other point from \(\nu\), proving both inclusions in this hull equality. The positive Fourier transform of the scalar function \(e^{i\nu t}\), regarded as a distribution, is instead supported at \(-\nu\). These are different uses of the transform. Since \(|e^{i\nu(t-is)}|=e^{\nu s}\), boundedness below the real axis selects \(\nu\leq0\) in the filter spectrum.

Here is a proof of the required half-plane criterion, relative to the exact Fourier, integration and local ideal inputs of OA-FLOW's spectral calculus. Their proofs are not given here. The general flow theorem is treated in OA-FLOW; this is its local application with the convention made explicit.

A bounded lower extension excludes positive frequencies. Suppose \(F(t)=\gamma_t(x)\) extends holomorphically below the real axis, with \(\|F(z)\|\leq C\), and has its weak-star continuous boundary values. For \(f\in C_c^\infty(\mathbb R)\) supported in \((c,\infty)\), \(c>0\), set

This is entire. Two integrations by parts in \(p\), together with the undifferentiated estimate for \(|t|\leq1\), give

In particular \(b_f\in L^1(\mathbb R)\) and \(\mathcal F b_f=f\). For completeness, the latter inversion follows by inserting \(e^{-\eta t^2}\), applying Fubini and the Gaussian Fourier identity specified in OA-FLOW, and then letting \(\eta\downarrow0\). The resulting Gaussian approximate identity tends to \(f(p)\), while dominated convergence on the \(L^1\) function \(b_f\) removes the factor. Thus no distributional inversion or closed-set spectral synthesis is being assumed in this test-function argument.

For a normal functional \(\omega\), shift the contour of \(b_f(z)\omega(F(z))\) down by \(s\). Its vertical integrals vanish: on any fixed horizontal strip the two integrations by parts give \(O(|\operatorname{Re}z|^{-2})\), and \(F\) is bounded. If the boundary is only weak-star continuous, first start at depth \(\eta>0\), then pass to \(\eta\downarrow0\) by the same integrable bound. Consequently

Normal functionals separate the algebra, so every such filter kills \(x\). At each \(p>0\) choose such an \(f\) with \(f(p)\ne0\). It belongs to the annihilator ideal and excludes \(p\) from its hull. Hence \(\operatorname{Sp}_\gamma(x)\subseteq(-\infty,0]\).

Negative spectrum supplies the contractive extension. Now let \(\gamma\) be a sigma-strong-star continuous automorphism flow on a von Neumann algebra and assume that spectrum inclusion. For \(s>0\), put

The mass identity is the elementary arctangent integral. For \(p>0\), integrate \(e^{ipz}s/(\pi(z^2+s^2))\) on the rectangle with vertices \(-R,R,R+iR,-R+iR\), where \(R>s\). The two vertical integrals are \(O(R^{-1})\); the upper integral is \(O(R^{-1}e^{-pR})\). The only pole inside is \(is\), with coefficient \(e^{-ps}/(2\pi i)\). Subtract that simple-pole term and apply the rectangle Cauchy formula to obtain the real integral \(e^{-ps}\). For \(p<0\) use the lower rectangle, with clockwise orientation and pole \(-is\), giving \(e^{sp}\). This proves (UW.54) using the scalar rectangle Cauchy input, without importing an arbitrary-cycle residue theorem. Both \(\partial_sP_s\) and \(P_s'\) are in \(L^1\), locally continuously in \(s>0\), as their explicit rational formulas show.

Define the weak-star Poisson integral

Positivity and unit mass give \(\|F(t-is)\|\leq\|x\|\). We verify holomorphy without assuming spectral synthesis at zero. For \(\delta>0\), use the dual Banach action \(\beta_t^\delta=e^{-i\delta t}\gamma_t\). It is isometric and has continuous predual orbits; it need not be an automorphism action. The frequency-origin identity in OA-FLOW gives \(\operatorname{Sp}_{\beta^\delta}(x) =\operatorname{Sp}_\gamma(x)-\delta\subseteq(-\infty,-\delta]\). This also follows directly from (UW.50) by translating the Fourier function. Let \(F_\delta\) be (UW.55) with \(\gamma\) replaced by \(\beta^\delta\). Writing its kernel as \(P_s(t-v)\) makes both derivatives norm derivatives: the difference quotients converge in \(L^1\), and the action is bounded. The kernel for \(\partial_sF_\delta+i\partial_tF_\delta\), written as a filter of \(\beta^\delta_t(x)\), is

Its Fourier support is contained in \([0,\infty)\), disjoint from \(\operatorname{Sp}_{\beta^\delta}(x)\). OA-FLOW's filter-support inclusion (S16) and its empty-spectrum criterion therefore make this filter zero. Thus \(\partial_sF_\delta+i\partial_tF_\delta=0\), the Cauchy--Riemann equation for \(z=t-is\). Continuous norm derivatives prove norm holomorphy. Put \(E_\delta=F_\delta(t-is)-F(t-is)\). The integral and (UW.54), with Cauchy--Schwarz for the probability measure \(P_s(r)\,dr\), give

This tends to zero uniformly on every compact subset of the lower half-plane. The norm limit is holomorphic and retains its norm bound.

In a faithful normal representation, the vector integral corresponding to (UW.55) exists: each continuous vector orbit on \(\mathbb R\) is separable and bounded. Put \(Y=F(t-is)-\gamma_{t_0}(x)\) and \(D(r)=\gamma_{t-r}(x)-\gamma_{t_0}(x)\). Jensen's inequality gives

As \((t,s)\to(t_0,0)\), split the integral at a small fixed \(|r|\). Strong continuity controls the part near zero; the remaining Poisson mass tends to zero, with integrand at most \(4\|x\|^2\|\xi\|^2\). The same argument for \(x^*\) gives strong-star continuity. Uniform boundedness makes this sigma-strong-star continuity as well: each summable family of vector seminorms is handled by a finite head and its uniformly bounded tail. Thus (UW.55) has the required boundary regularity, without a separability assumption on the algebra or its representation.

Apply both directions to \(\gamma=\sigma^\Phi\) and \(x=1\otimes e_{12}\). Every integrand in (UW.55) remains in the 12-corner by (UW.9), so the extension has the form \(u_z\otimes e_{12}\). The corner embedding is isometric and preserves the boundary topology. We obtain the corrected equivalence

Taking adjoints instead gives the positive half-line for \(e_{21}\). Indeed the filter identity for \(x^*\) replaces \(f(p)\) by \(\overline{f(-p)}\), so its annihilator hull is reflected. The earlier positive half-line for \(e_{12}\) was incompatible with the unchanged OA-FLOW convention. This identifies a local error; it makes no claim about an error in the source book's conventions.

A complete scalar sign test. Take \(M=\mathbb C\), \(\psi(r)=r\) and \(\varphi(r)=r/2\) for \(r\geq0\). These are faithful normal finite weights. The balanced density on \(M_2(\mathbb C)\) is \(\operatorname{diag}(1/2,1)\) relative to its usual trace. Hence

Thus \(\varphi\preceq_\infty\psi\), while the earlier positive half-line condition fails. The adjoint \(e_{21}\) has frequency \(\log2\). For equal weights the frequency is zero, retained by both closed half-lines. The zero vector has empty spectrum, as in OA-FLOW.

The extended uniform distance

Scaling a weight changes the cocycle by

Define

This is an extended metric. Multiplying both weights in a comparison by the same positive scalar leaves its cocycle unchanged. Hence the two inequalities in (UW.12), after reciprocal rescaling, also give

which proves symmetry. If \(a\) and \(b\) are admissible for \((\varphi,\psi)\) and \((\psi,\chi)\), respectively, transitivity gives

Taking infima proves the triangle inequality. If \(d(\varphi,\psi)=0\), (UW.8) gives, for every \(a>0\),

Letting \(a\downarrow0\) at each positive element, including an infinite value, gives \(\varphi=\psi\).

If \(d(\varphi,\psi)<a\), the cocycle

extends to an entire \(M\)-valued function satisfying

Indeed, the two comparisons in (UW.12) give compatible extensions in the lower and upper half-planes after applying (UW.11); they glue across the real axis. The scalar factor in (UW.11) gives exactly the exponential in (UW.17). The topology defined by \(d\) is the uniform topology on \(\mathcal W_0(M)\).

Trace densities and the factor-of-two correction

Let \(M\) be semifinite, let \(\tau\) be a faithful normal semifinite trace, and let \(h,k\) be positive injective self-adjoint operators affiliated with \(M\). Put

The density convention is the one in PT-08, so that

For \(\lambda>0\), rescale time by \(2\lambda\). The function

has a contractive half-strip extension exactly when the cocycle in (UW.19) has a contractive strip of width \(\lambda\). Equations (UW.8) and PT-08's order-preserving density correspondence therefore prove

The printed XII.5.6 omits the factor \(2\) in (UW.21). Its formula cannot be reconciled with the half-strip identity (UW.8). A concrete obstruction occurs in \(M_2(\mathbb C)\). Let

Then \(k-h\) is the positive rank-one matrix with every entry equal to one, but

Thus \(h\leq k\) while \(h^2\nleq k^2\). At \(\lambda=1\), the printed criterion would accept this pair, whereas the defining strip and (UW.8) correctly reject it.

The infinite-width conclusion printed after that formula remains valid. If \(p(s)=1_{[0,s]}(h)\) and \(q(s)=1_{[0,s]}(k)\), then

Only the implication from integer powers to spectral projections needs care. If \(\xi\in q(s)H\) and \(\mu>s\), the inequality with exponent \(2n\) gives

The spectral theorem then puts \(\xi\) in \(p(s)H\), so \(q(s)\leq p(s)\). Conversely, this projection order gives \(1-p(s)\leq1-q(s)\). The layer-cake formula for quadratic forms yields, for every \(r>0\),

The equality and inequality allow the value \(+\infty\). This proves all of (UW.24), and (UW.21) converts the all-real-power condition back to \(\preceq_\infty\).

Differentiable cocycles are bounded logarithmic perturbations

Keep the trace setting and put

The map \(t\mapsto u_t\) is sigma-strong-star differentiable if and only if

for a bounded self-adjoint \(a\in M\).

Suppose first that the derivative at zero exists. Differentiating \(u_t^*u_t=1\) shows that \(u'_0=ia\) for a bounded self-adjoint \(a\). For \(\xi\in D(H)\), the identity

has a strong derivative at zero. Stone's exact derivative-domain theorem SG-04 gives \(\xi\in D(K)\) and \(K\xi=(H+a)\xi\). Applying the same argument to \(e^{itH}=u_t^*e^{itK}\) gives the reverse domain inclusion. This proves (UW.28).

Conversely, a bounded self-adjoint perturbation preserves self-adjointness and the domain. The cocycle identity and the derivative at zero give the strong-star differential equation

and hence

The integral is sigma-strong-star. Iteration gives the time-ordered Dyson series; its \(n\)-th term has norm at most \(\|a\|^n|t|^n/n!\). It therefore converges in norm on compact time intervals and verifies (UW.30), proving the converse without differentiating an unbounded product formally.

Uniform distance bounds the perturbation spectrum

Let \(\varphi,\psi\in\mathcal W_0(M)\) and suppose

Write \(u_t=[D\psi:D\varphi]_t\), as in (UW.16). The entire extension in UW-04 is norm differentiable. Set \(a=-iu'_0\in M\); differentiation of \(u_t^*u_t=1\) at zero gives \(a=a^*\). Differentiating the cocycle identity \(u_{t+s}=u_t\sigma_t^\varphi(u_s)\) in \(s\) at zero gives (UW.30). In the trace-density setting of UW-06, its domain argument identifies \(a=K-H\) on \(D(H)=D(K)\). For arbitrary weights, the bounded cocycle derivative defines \(a\) without introducing trace densities. For every \(\varepsilon>0\), put \(b=\alpha+\varepsilon\). Equation (UW.17) gives

The differential equation supplies the actual modular orbit

The functions \(u_t\) and \(u'_t\) are cocycle functions; they have not been identified as orbits of fixed algebra elements. Therefore the fixed-element spectral-product theorem does not justify a product-support step here. We instead construct the entire continuation of (UW.34) and prove its Fourier-filter support directly.

Put \(u^\sharp(z)=u(\bar z)^*\). This is entire: if \(u(w)=\sum_n c_n(w-\bar z_0)^n\) near \(\bar z_0\), then \(u^\sharp(z)=\sum_n c_n^*(z-z_0)^n\) near \(z_0\). Its norm has the same bound as \(u\). The norm Cauchy estimate on the circle of radius one about \(z\) gives

Consequently

Both factors are norm holomorphic and multiplication is a bounded bilinear map, so \(G\) is entire. The reflection in \(u^\sharp\) is essential; \(z\mapsto u(z)^*\) alone would not be holomorphic.

Take \(f\in C_c^\infty(\mathbb R)\) supported in \((c,\infty)\), where \(c>2b\), and use its inverse Fourier function \(b_f\) from (UW.51). The inversion and integrability proved in UW-03 make \(\sigma_f^\varphi(a)\) exactly the filter in (UW.50). For a normal functional \(\omega\), the rectangle contour shift gives

For fixed \(s\), the vertical sides vanish because \(b_f\) is \(O(|t|^{-2})\) uniformly on that strip and \(G\) is bounded there by (UW.61). All horizontal integrals are absolutely convergent. Equations (UW.52) and (UW.61) give

Thus every such filter annihilates \(a\). For \(f\) supported in \((-\infty,-c)\), shift upwards instead: \(b_f(t+is)\) has the same exponential decay, and (UW.61) is symmetric in the two half-planes. This proves annihilation for the negative tests too.

At any point outside \([-2b,2b]\), choose a smooth compactly supported test function nonzero there and supported entirely on the corresponding side, with some \(c>2b\). It lies in the annihilator ideal, so the point is absent from the OA-FLOW hull. Hence \(\operatorname{Sp}_{\sigma^\varphi}(a)\subseteq[-2b,2b]\). Intersecting these closed intervals over all \(\varepsilon>0\) proves

The argument proves the claimed support at the filter definition, without a product of distributions or a spectral-synthesis assumption. Reflecting the Fourier sign leaves this symmetric interval unchanged. The case \(\alpha=0\) is included by the same intersection (and the metric gives \(\varphi=\psi\), hence \(a=0\)). The factor two in (UW.35) is present in the source and is distinct from the missing factor two corrected in (UW.21).

Entire functions of small exponential type vary little

Let \(E\) be a complex Banach space. Given \(\varepsilon,R>0\), there is \(\delta>0\) such that every entire \(f:\mathbb C\to E\) satisfying

also satisfies

First take scalar-valued entire functions. For \(c\geq0\), let \(\mathcal H_c\) denote those satisfying

For \(0<c\leq1\), these functions are uniformly bounded on each compact set. Cauchy's estimate makes them equicontinuous on every smaller concentric disc. Arzela--Ascoli on discs of integer radius, followed by a diagonal subsequence, gives locally uniform convergence; the limit is entire by the local Cauchy formula and retains the defining bound. The compact-open topology is metrized by the weighted sum of the suprema on those discs. Thus \(\mathcal H_c\) is compact. The sets decrease as \(c\downarrow0\), and their intersection consists exactly of the constants of modulus at most one. Indeed, if \(g\) belongs to every \(\mathcal H_c\), the Cauchy estimate on the circle of radius \(L\) about \(z_0\), used with \(c=L^{-2}\), gives

Letting \(L\to\infty\) gives \(g'(z_0)=0\). Thus \(g\) is constant, and its real-axis bound is at most one.

Let

This is an open neighborhood of \(\mathcal H_0\). If no \(\mathcal H_\delta\) lay in \(\mathcal U\), choose \(g_n\in\mathcal H_{1/n}\setminus\mathcal U\). A subsequence converges on compact sets to an element of every \(\mathcal H_c\), hence to an element of \(\mathcal H_0\subset\mathcal U\), contradicting openness. Thus some \(\delta\) works for scalar functions.

For Banach-valued \(f\), compose with every \(\ell\in E^*\) of norm at most one. The scalar result gives

Hahn--Banach norms the vector difference, proving (UW.37). This proof also shows that \(\delta\) depends only on \(\varepsilon\) and \(R\), not on \(E\).

Completeness and continuity

The extended metric space \((\mathcal W_0(M),d)\) is complete. Let \((\varphi_n)\) be \(d\)-Cauchy. Choose a tail index \(N\) and set \(\rho=\varphi_N\). Every tail weight has finite distance from \(\rho\), and these distances are uniformly bounded. Hence

is entire and uniformly bounded on each compact subset of \(\mathbb C\).

Fix \(R>0\). If \(d(\varphi_n,\varphi_m)<\delta\), write

UW-08 makes \(W_{nm}(z)\) uniformly close to one for \(|z|\leq R\) once \(\delta\) is small. Analytic continuation of the chain rule gives

The common compact bound on \(U_m\) now shows that \((U_n)\) is uniformly Cauchy on \(|z|\leq R\). Its locally uniform norm limit \(V\) is entire. On the real axis it is a strongly continuous unitary \(\sigma^\rho\)-cocycle: unitarity and the cocycle law pass through the norm limit. The reconstruction theorem CX-10 gives a unique \(\psi\in\mathcal W_0(M)\) with

For \(\varepsilon>0\), choose \(n\) so late that \(d(\varphi_m,\varphi_n)<\varepsilon\) for every later \(m\). The functions \([D\varphi_m:D\varphi_n]_z\) have the common bound \(e^{\varepsilon|\operatorname{Im}z|}\). Their locally uniform limit is the analytic continuation of \([D\psi:D\varphi_n]_t\), by (UW.43)--(UW.44). The same bound survives the limit, so

Thus \(\varphi_n\to\psi\) in the uniform topology. Choosing the fixed tail reference \(\rho\) explicitly avoids assuming in advance that the whole sequence lies in one finite-distance component.

For every \(x\in M_+\), the evaluation map

is continuous into \([0,\infty]\) with its order topology. Indeed, \(d(\varphi,\psi)<a\) and (UW.8) give

This squeeze also handles the value \(+\infty\).

Finally, for faithful normal states \(\varphi,\psi\),

If \(d(\varphi,\psi)<a\), (UW.47) in both directions gives, for \(0\leq x\leq1\),

The self-adjoint functional \(\varphi-\psi\) vanishes at one, so its norm is twice the supremum in (UW.49). If \(d<1/2\), let \(a\downarrow d\) and use \(e^a-1\leq2a\) on this interval. If \(d\geq1/2\), the trivial state bound \(\|\varphi-\psi\|\leq2\) gives the same conclusion. This proves (UW.48) for every finite \(d\); it is automatic when \(d=\infty\).

Scope and ownership

UW-01--09 cover Takesaki II XII.5, Definition 5.1 through Lemma 5.8, with the density exponent corrected in (UW.21). The half-strip domination theorem is the explicit OA-MOD prerequisite OA-MOD-UW-DEP-DOMINATION. The general Arveson/Paley--Wiener theorem is the explicit OA-FLOW-owned prerequisite OA-MOD-UW-DEP-ARVESON-PW. Crossed products, dual actions, action cocycles, flow classification, and the later action of inner automorphisms remain with OA-FLOW.

References

• M. Takesaki, Theory of Operator Algebras II, Springer, 2003, Chapter XII, Section 5, printed pages 421--425.
• M. Takesaki, Theory of Operator Algebras II, Chapter VIII, Section 3, Theorem 3.17.