Original text: CC0 1.0. Prerequisite proofs and component terms.

Modular time on a crossed-product coefficient algebra

OA-FLOW.DMO.SETTING — Fixed conventions and available inputs

Let $G$ be an arbitrary locally compact Hausdorff group with fixed left Haar measure, and let $\alpha:G\to\operatorname{Aut}(M)$ be point-ultraweakly continuous. Fix an n.s.f. weight $\varphi$, with the standard GNS representation on $H_\varphi$. There is no assumption of separability, second countability, sigma compactness, unimodularity, boundedness of the weight, or invariance of $\varphi$.

Use $\mathcal K$, $\mathfrak b_\varphi$, $B_\varphi$, $\widetilde\Lambda_\varphi$, and $\mathcal A_\varphi$ from lesson 09. Thus $\mathcal K$ has right-coefficient convolution, $\mathfrak b_\varphi=\operatorname{span}(\mathcal K\cdot\mathfrak n_\varphi)$, $B_\varphi=\mathfrak b_\varphi\cap\mathfrak b_\varphi^\sharp$, and $\mathcal A_\varphi=\widetilde\Lambda_\varphi(B_\varphi)$ is dense in $\mathcal H=L^2(G,H_\varphi)$. The defining operations are

The modular function satisfies $\int h(rs)\,dr=\Delta_G(s)^{-1}\int h(r)\,dr$. We write $\sigma_t=\sigma_t^\varphi$, $\psi_s=\varphi\circ\alpha_s$, and

The jointly sigma-strong-star continuous dependence of $c_t(s)$ on $(s,t)$ is the theorem OA-FLOW.DW.COCYCLECONT in lesson 10. Its group/time identities and exact factor order are proved in OA-FLOW.AWC.LAWS. These results retain their stated unresolved prerequisites; merely using them does not admit those prerequisites.

OA-FLOW.DMO.IMPORT.MODULAR — Exact OA-MOD operator inputs

In addition to OA-FLOW.DW.IMPORT.GNS and OA-FLOW.DW.IMPORT.TOMITA, the following are the precise general modular inputs, owned by OA-MOD and presently specified imports.

The modular automorphisms form a continuous action in the predual topology, preserve $\varphi$ and $\mathfrak n_\varphi$, and obey

For any n.s.f. $\psi$, the relative modular operator on this standard Hilbert space is positive, self-adjoint and nonsingular, with

The relative Tomita operator has the domain inclusion, standard conjugation and transport formulas specified in (D23)–(D24) of lesson 09. In particular, its positive half-power is the one appearing in (M4). The standard implementing representation $U_s$ of $\alpha_s$ is strongly continuous and commutes with the standard conjugation $J$.

Naturality and the Connes derivative rules used in lesson 10 give, for all $s,r,t,u$,

These identities include their normalizations at the identity and at time zero. They do not assert that the pullback coefficient $s\mapsto c_t(s)$ is an ordinary left $\alpha$-cocycle.

OA-FLOW.DMO.IMPORT.SPECTRAL — Exact scalar and Hilbert-space calculus

The harmonic-analysis input is Stone's theorem and the spectral calculus on arbitrary Hilbert spaces, including closed Borel functions of a self-adjoint operator, bounded spectral approximations, and uniqueness of the polar decomposition of a densely defined closed antilinear operator. These are refinements of IMP.HARMONIC and are used here without proof.

We also require scalar Fourier inversion for $k\in C_c^\infty(\mathbb R)$, in the normalization

Here $\widehat k\in L^1(\mathbb R)$. The spectral theorem then gives

as a strong integral for self-adjoint $P$; its norm is at most $\|k\|_\infty$. In passing from (M6) to (M7), the scalar spectral measures are finite and the Fourier kernel is integrable. This is a one-variable spectral-calculus input. No theorem on decomposable fields of operators is included among the assumptions.

OA-FLOW.DW.COEFFICIENTFLOW — The modular-time action on coefficients

For $f\in\mathcal K$, define

Theorem. The maps $R_t$ form a one-parameter group of star automorphisms of the convolution algebra $\mathcal K$. They preserve supports and the coefficient $L^1$ norm. For $a\in M$,

Both $\mathfrak b_\varphi$ and $B_\varphi$ are invariant under every $R_t$.

Proof. Joint continuity of $c_t(s)$, bounded strong-star continuity of the modular action, and bounded-product continuity imply that $(s,t)\mapsto(R_tf)(s)$ is jointly strong-star continuous. Each multiplier $\Delta_G(s)^{it}c_t(s)$ is unitary, and $\sigma_t$ is isometric. Thus

which proves support preservation, membership in $\mathcal K$, and preservation of the $L^1$ norm. The time cocycle identity in (M5) gives $R_tR_u=R_{t+u}$; the normalizations give $R_0=1$, hence $R_t^{-1}=R_{-t}$.

We check convolution without commuting any coefficients. Fix $t$, write $C(s)=c_t(s)$ and $d(s)=\Delta_G(s)^{it}$. Since $d(rs)d(s^{-1})=d(r)$, the convolution integrand is

The second line uses both the group identity and the inverse identity in (M5). For the third line, the final identity in (M5) is equivalently

Integrating (M10), and using normality of $\sigma_t$ to pass it through the compact coefficient integral, proves $(R_tf)*(R_tg)=R_t(f*g)$.

For the involution, put $x=f(r^{-1})^*$. Direct substitution gives

The third line is again modular naturality in (M5). Complex conjugation of the scalar factor in $(R_tf)(r^{-1})$ has changed it to $d(r)$; the real factor $\Delta_G(r)^{-1}$ remains from the coefficient involution.

The right identity in (M9) follows by applying $\sigma_t$ to the product $f(s)a$. For the left identity, (M5) gives

which gives the required equality pointwise. Since $\sigma_t(\mathfrak n_\varphi)=\mathfrak n_\varphi$, the right coefficient identity takes every finite sum in $\mathfrak b_\varphi$ into that same domain. Apply the inverse $R_{-t}$ to obtain equality. Finally (M11) gives $R_t(B_\varphi)=B_\varphi$. $\square$

For fixed $s$, the map $a\mapsto c_t(s)\sigma_t(a)$ is usually not a unital star automorphism of $M$. The star automorphism in the theorem acts on the entire twisted convolution algebra. The group identities in (M5) are what make its multiplicativity possible.

OA-FLOW.DW.MODULARUNITARIES — Constructing a unitary group on the section space

Set

Each $A_s$ is positive, self-adjoint and nonsingular. Formula (M4) and the joint-continuity theorem give joint strong continuity of $(s,t)\mapsto q_t(s)$, and of its adjoint.

Proposition. There is a strongly continuous unitary group $(Q_t)$ on $\mathcal H$ whose action is

There is a unique positive nonsingular self-adjoint $\widetilde\Delta_\varphi$ such that $Q_t=\widetilde\Delta_\varphi^{it}$.

Proof. Begin with $\xi\in C_c(G,H_\varphi)$. The right side of (M13) is continuous with the same compact support, and pointwise unitarity gives $\|Q_t\xi\|_2=\|\xi\|_2$. Thus it extends by density to an isometry on $\mathcal H$. The pointwise group law for $A_s^{it}$, first on these continuous vectors, gives $Q_tQ_u=Q_{t+u}$ and inverse $Q_{-t}$; hence the extensions are unitary.

On a compact support $K$, the map $(s,t)\mapsto q_t(s)\xi(s)$ is norm continuous. A finite-subcover argument shows that its difference from $\xi(s)$ tends uniformly to zero in $s\in K$ as $t\to0$. Since $K$ has finite Haar measure, $Q_t\xi\to\xi$ in $L^2$. For any $\eta\in\mathcal H$, approximate it by such a $\xi$ and use

This proves strong continuity. For a general $\xi$, formula (M13) holds almost everywhere: choose a sequence of compact-support continuous approximants, and pass to subsequences converging almost everywhere both before and after applying the bounded operator $Q_t$. Pointwise unitarity identifies the second limit with $q_t(s)\xi(s)$. This also establishes the required strong measurability of the resulting section. The assertion is for each fixed $t$ and vector; no common null set for all vectors and all times is asserted.

By Stone's theorem there is a self-adjoint $P$ with $Q_t=e^{itP}$. Put $\widetilde\Delta_\varphi=e^P$. The spectral function $e^p$ is strictly positive, so its operator has zero kernel, even if its inverse is unbounded. Its imaginary powers are $e^{itP}$. Conversely, the logarithm of any positive nonsingular operator with those imaginary powers is the same Stone generator, giving uniqueness. $\square$

The construction begins on compact-support vector sections. Strong continuity on an entire non-sigma-compact group was not treated as automatic global Bochner measurability of every vector orbit.

OA-FLOW.DMO.FUNCTIONALCALCULUS — Smooth bounded functions act pointwise

Write $P_s=\log A_s$, so $q_t(s)=e^{itP_s}$. We need only smooth compactly supported spectral functions to obtain all real-power domains.

Lemma. For every $k\in C_c^\infty(\mathbb R)$ and $\xi\in\mathcal H$,

In particular the right side represents a strongly measurable $L^2$ section. Both sides have norm at most $\|k\|_\infty\|\xi\|_2$.

Proof. First suppose $\xi$ is continuous with compact support $K$. By (M7),

The integrand is strongly measurable as a Hilbert-space function of $t$, and its norm is bounded by $|\widehat k(t)|\|\xi\|_2$, an integrable scalar function. Pointwise on $G$, the candidate integral is

This equality is the individual spectral theorem for $P_s$. To compare (M15) and (M16), restrict first to a bounded time interval. The vector integrands are jointly continuous on that interval times $K$; they have separable range there because their range is norm compact. On this product, Lebesgue measure is finite and Haar measure of $K$ is finite. Bochner Fubini, or uniformly convergent vector Riemann sums on the time interval followed by compact-support integration, identifies the two integrals in $L^2(G,H_\varphi)$.

Let the time interval expand. The omitted tail in the pointwise integral is bounded by

and hence its $L^2$ norm is bounded by the same scalar tail times $\|\xi\|_2$. The tail in (M15) has that bound as well. This proves (M14) for compact-support continuous vectors. It also shows directly that its right side is a continuous compact-support vector function in this case.

For an arbitrary $\xi$, choose such vectors $\xi_n\to\xi$ in $L^2$. Since $\|k(P_s)\|\leq\|k\|_\infty$ for every $s$, the pointwise multiplication formula is uniformly bounded. Choose subsequences with $\xi_n(s)\to\xi(s)$ and $(k(P)\xi_n)(s)\to(k(P)\xi)(s)$ outside one null set. The bounded pointwise operators identify the two limits, proving (M14). The left side already supplies a strongly measurable $L^2$ representative. No countable basis for $H_\varphi$, and no general operator-field measurability theorem, has been used. $\square$

OA-FLOW.DW.REALPOWERDOMAINS — Every real exponent, with its full domain

Theorem. Let $r\in\mathbb R$. A vector $\xi\in\mathcal H$ belongs to $\operatorname{Dom}(\widetilde\Delta_\varphi^r)$ if and only if

On this domain,

The same statement includes negative $r$. Nonsingularity does not make the negative powers bounded.

Proof. Choose a smooth function $\chi:\mathbb R\to[0,1]$ supported in $[-2,2]$ and equal to $1$ on $[-1,1]$. Put

Both are smooth and compactly supported. The spectral theorem gives $\chi_n(P)\to1$ and $\chi_n(P_s)\to1$ strongly, and

where the operators in (M20) are everywhere defined and bounded for each fixed $n$. No bound uniform in $n$ is needed for $h_n$.

Suppose first that $\xi\in\operatorname{Dom}(e^{rP})$, with $\zeta=e^{rP}\xi$. The spectral calculus gives

in $L^2$. By (M14), these bounded operators act pointwise as their counterparts for $P_s$. Take a subsequence along which $(h_n(P)\xi)(s)\to\zeta(s)$ almost everywhere, and remove the countable collection of null sets from (M14). At each remaining point, $\chi_n(P_s)\xi(s)\to\xi(s)$ by the individual spectral theorem, while its image under the closed operator $e^{rP_s}$ tends to $\zeta(s)$. Closedness therefore yields

Since $e^{rP_s}=A_s^r=\Delta_G(s)^r\Delta_{\psi_s,\varphi}^r$, this proves (M17), including measurability and integrability of the image.

Conversely, suppose (M17) holds. Applying (M14) to $h_n$ and $\chi_n$, and commuting the cutoff with the individual spectral power on its domain, gives

almost everywhere. Hence $h_n(P)\xi=\chi_n(P)\eta\to\eta$ in $L^2$, while $\chi_n(P)\xi\to\xi$. Equation (M20) and closedness of $e^{rP}$ imply that $\xi$ is in its domain and that $e^{rP}\xi=\eta$. This proves the reverse inclusion and the operator formula. The norm equality in (M18) is then the definition of the section norm. $\square$

Strong measurability in (M17) is explicit so that the operator domain is well-typed. In fact it follows already from the pointwise domain condition in this setting: by (M14), each $h_n(P_s)\xi(s)$ has a strongly measurable representative; on the full-measure set of domain membership these vectors converge to $e^{rP_s}\xi(s)$. A countable almost-everywhere pointwise limit of strongly measurable Hilbert-space functions is strongly measurable. Thus one can equivalently use pointwise domain membership and the finiteness of the norm integral. There is no additional measurable-field hypothesis hidden in this reformulation.

OA-FLOW.DMO.INTERTWINING — Modular time preserves the common domain

Proposition. For every $f\in\mathfrak b_\varphi$ and every real $t$,

In particular $\widetilde\Delta_\varphi^{it}\mathcal A_\varphi=\mathcal A_\varphi$.

Proof. The coefficient-flow theorem proves $R_tf\in\mathfrak b_\varphi$. For each $s$, left multiplication by a bounded coefficient and (M3) give

This is (M23) by (M13). The invariance of $B_\varphi$ under $R_t$ gives the final assertion, including equality by applying $-t$. $\square$

Only pointwise weight-finite elements were used in the GNS formulas, and their membership was supplied by lesson 09 and $\varphi$-invariance of $\sigma_t$. No assertion about applying the GNS map to an arbitrary element of $M$ is made.

OA-FLOW.DW.INVARIANTCORE — The closed involution and its polar decomposition

Theorem. The space $\mathcal A_\varphi$ is a graph-norm core for $\widetilde\Delta_\varphi^{1/2}$, and

This is the polar decomposition of the closed coefficient involution. Consequently the modular operator and conjugation of the conditional left Hilbert algebra from lesson 09 are precisely $\widetilde\Delta_\varphi$ and $\mathcal J$.

Proof. The full domain theorem with $r=1/2$ identifies $T=\widetilde\Delta_\varphi^{1/2}$, including equality of domains. The pointwise relative Tomita computation of lesson 09 therefore gives

Lesson 09 proved that $\mathcal A_\varphi$ is dense. Equation (M23) supplies its invariance under every imaginary power. All the hypotheses of OA-FLOW.GRAPH.POWERS now hold, so $\mathcal A_\varphi$ is a core for the half-power. Applying the bounded isometry $1\oplus\mathcal J$ to the relevant graphs proves (M25).

The positive factor $\widetilde\Delta_\varphi^{1/2}$ is nonsingular and has dense range, and $\mathcal J$ is antiunitary. Thus (M25) is its polar decomposition, by the uniqueness statement in the operator-calculus import. The terminology of modular objects for the left Hilbert algebra refers to these polar factors; it does not yet invoke a dual-weight formula on arbitrary crossed-product positive elements. $\square$

The domain theorem covers every real exponent on its own maximal domain. The common coefficient algebra has been proved to lie in the half-power domain. It has not thereby been shown to lie in every higher-power domain; a solved problem below exhibits the distinction.

Editable source · Proof dependencies and component terms