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

Moving an action and a modular parameter together

OA-FLOW.AWC.SETTING — Objects, parameters, and topologies

Let $G$ be an arbitrary locally compact Hausdorff group with a chosen nonzero left Haar measure $dr$ . Let $M$ be a nonzero von Neumann algebra and let $\alpha:G\to\operatorname{Aut}(M)$ be a point-ultraweakly continuous action by normal automorphisms. Thus $s\mapsto\omega(\alpha_s(x))$ is continuous for every $x\in M$ and $\omega\in M_*$ . Fix an n.s.f. weight $\varphi$ on $M$ and set

c(s,t)=[D(\varphi\circ\alpha_s):D\varphi]_t\in\mathcal U(M),\qquad (s,t)\in G\times\mathbb R.

Composition is pullback of a weight: $(\varphi\circ\alpha_s)(x)=\varphi(\alpha_s(x))$ for $x\ge0$ . Consequently the group parameter appears with inverses in several naturality formulas. We use the action convention $\alpha_{sr}=\alpha_s\alpha_r$ throughout.

The $\sigma$ -strong topology on $M$ is generated by

p_\omega(x)=\omega(x^*x)^{1/2},\qquad\omega\in M_*^+.

The $\sigma$ -strong-* topology also uses $p_\omega(x^*)$ . Joint continuity will mean continuity on the product topological space $G\times\mathbb R$ , equivalently convergence for every convergent net of pairs. There is no assumption of separability of $M$ , second countability or $\sigma$ -compactness of $G$ , unimodularity, a faithful normal state, or invariance of $\varphi$ .

If the zero algebra is included in the convention for von Neumann algebras, its unitary group has one element and the assertion is immediate. The nonzero assumption above only removes that vacuous case from the construction.

OA-FLOW.AWC.IMPORTS — Exact prerequisites and their ownership

The results stated below are used as precise assumptions; their proofs are not given here.

OA-FLOW.AWC.IMPORT.ACTIONTOPOLOGY — Action topology, owned by OA-FLOW

The action-topology portion of IMP.CP.REGULAR identifies point-ultraweak continuity of a group action with norm continuity of each predual orbit. This equivalence is used only to apply the standard-form continuity theorem. This unit does not assume operator-norm continuity of the orbit $s\mapsto\alpha_s(x)$ .

OA-FLOW.AWC.IMPORT.STANDARD — Standard form, owned by OA-MOD

Choose a faithful normal standard representation of $M$ on a Hilbert space $H$ , with natural cone. Every $\omega\in M_*^+$ has a representing cone vector $\xi_\omega$ satisfying $\omega(x)=\langle x\xi_\omega,\xi_\omega\rangle$ . Each normal automorphism has a canonical unitary implementer, and these implementers preserve composition. The standard-implementation map is continuous from the predual topology on automorphisms to the strong topology on unitaries. Hence our action has a strongly continuous unitary representation $U:G\to\mathcal U(H)$ with

U_s x U_s^*=\alpha_s(x),\qquad U_{sr}=U_sU_r.

The cone vector represents a bounded normal positive functional, not the possibly infinite weight $\varphi$ . No vector representing $\varphi$ as a bounded functional is assumed.

OA-FLOW.AWC.IMPORT.WEIGHTS — Weight calculus, owned by OA-MOD

Pullback by a normal automorphism and tensor product preserve the class of n.s.f. weights. If $\eta,\zeta,\rho$ are n.s.f. weights on one algebra, their derivatives are $\sigma$ -strongly continuous unitary families with

[D\eta:D\zeta]_t[D\zeta:D\rho]_t=[D\eta:D\rho]_t,\qquad [D\eta:D\eta]_t=1.

Their time cocycle law and modular implementation are

[D\eta:D\zeta]_{t+u}=[D\eta:D\zeta]_t\,\sigma_t^\zeta([D\eta:D\zeta]_u),

\sigma_t^\eta=\operatorname{Ad}([D\eta:D\zeta]_t)\sigma_t^\zeta.

For a normal automorphism $\beta$ of that algebra, naturality means

[D(\eta\circ\beta):D(\zeta\circ\beta)]_t=\beta^{-1}([D\eta:D\zeta]_t),\qquad \sigma_t^{\eta\circ\beta}=\beta^{-1}\sigma_t^\eta\beta.

For n.s.f. weights on two algebras, the tensor identities required are

[D(\eta_1\otimes\eta_2):D(\zeta_1\otimes\zeta_2)]_t =[D\eta_1:D\zeta_1]_t\otimes[D\eta_2:D\zeta_2]_t,

(\eta_1\otimes\eta_2)\circ(\beta_1\otimes\beta_2) =(\eta_1\circ\beta_1)\otimes(\eta_2\circ\beta_2).

These are identities of weights on the entire positive cone, including infinite values. Equality on elementary positive tensors is not used as an unsupported uniqueness theorem for weights. The examples use the scaling formula $[D(a\eta):D\eta]_t=a^{it}1$ , $a>0$ . For positive invertible matrices $H,K$ and $\varphi_H(x)=\operatorname{Tr}(Hx)$ they also use the exact finite-dimensional formulas

\sigma_t^{\varphi_H}(x)=H^{it}xH^{-it},\qquad [D\varphi_K:D\varphi_H]_t=K^{it}H^{-it}.

The general derivative and standard-form theorems remain sibling imports, rather than being reproved through informal calculations with unbounded densities.

OA-FLOW.AWC.IMPORT.HAARTENSOR — Haar and spatial tensor foundations

We use the Haar multiplication von Neumann algebra $D=L^\infty(G)$ on $L^2(G,dr)$ , including its full localizable interpretation when Haar measure is not $\sigma$ -finite. It is maximal abelian and is generated as a von Neumann algebra by the multiplication operators from $C_c(G)$ . Its Haar integration weight

\mu(f)=\int_G f(r)\,dr\qquad(f\in D_+)

is n.s.f. We use density of $C_c(G)$ in $L^2(G)$ , finiteness of Haar measure on compact sets, and left invariance. For arbitrary Hilbert spaces and von Neumann algebras we use the spatial tensor construction, its normality, and the tensor commutant identity. In particular, on $H\otimes L^2(G)$ ,

(M'\,\overline\otimes\,D)'=M\,\overline\otimes\,D.

These precise parts of IMP.HARMONIC remain foundational imports. No measurable-field decomposition theorem, countable field basis, or crossed-product dual-weight theorem is needed in this proof.

OA-FLOW.AWC.VECTORCONT — Two elementary continuity estimates

We record the topology arguments so that no appeal to separate continuity is hidden at the end.

Lemma. Suppose $A_i\to A$ strongly and $\sup_i\|A_i\|\le C$ , while $\xi_i\to\xi$ in norm. Then $A_i\xi_i\to A\xi$ in norm. Products of a fixed finite number of strongly convergent, uniformly bounded operator nets therefore converge strongly. On the unitary group, inversion is strongly continuous.

Proof. The first assertion follows from

\|A_i\xi_i-A\xi\|\le C\|\xi_i-\xi\|+\|(A_i-A)\xi\|.

Apply this estimate repeatedly to obtain the product assertion. If $V_i\to V$ strongly and all operators, including the limit, are unitary, then

\|(V_i^*-V^*)\xi\|=\|\xi-V_iV^*\xi\|\longrightarrow0.

This proves the last assertion for nets, not just sequences. $\square$

Corollary. Let $s\mapsto V_s$ be a strongly continuous unitary representation of any topological group, and let $t\mapsto v_t$ be a strongly continuous unitary curve. Then

(s,t)\longmapsto V_s^*v_tV_s v_t^*

is jointly strong-* continuous.

Proof. For a convergent net $(s_i,t_i)\to(s,t)$ , each of its four factors converges strongly, has norm one, and has strongly convergent adjoint by the lemma. Apply the finite-product assertion to the product and to its adjoint. $\square$

OA-FLOW.AWC.TRANSLATION — Left translation on the extra coordinate

Define

(L_s f)(r)=f(s^{-1}r),\qquad (\ell_s h)(r)=h(s^{-1}r),

where $L_s$ acts on $L^2(G)$ and $\ell_s$ acts on $D=L^\infty(G)$ . Left invariance gives $\|L_sf\|_2=\|f\|_2$ , $L_sL_r=L_{sr}$ , and $L_s^{-1}=L_{s^{-1}}$ . It also gives

\mu\circ\ell_s=\mu.

There is no modular-function factor: the integration variable was translated on the left. This assertion remains true for nonunimodular groups.

Lemma. The representation $s\mapsto L_s$ is strongly continuous for an arbitrary locally compact $G$ .

Proof. It is enough to check continuity at the identity. Take $f\in C_c(G)$ and a relatively compact identity neighborhood $V$ . For $s\in V$ , the supports of $L_sf$ and $f$ lie in the compact set

K=\overline V\operatorname{supp}(f)\cup\operatorname{supp}(f).

Joint continuity of $(s,r)\mapsto f(s^{-1}r)-f(r)$ and compactness of $K$ give

\sup_{r\in K}|f(s^{-1}r)-f(r)|\longrightarrow0\quad(s\to e).

For completeness, for a prescribed error choose, at each $(e,r)$ , a product neighborhood where the error is smaller; take a finite subcover in the $r$ variable and intersect the corresponding identity neighborhoods in the $s$ variable. Therefore

\|L_sf-f\|_2\le\mu(K)^{1/2}\sup_{r\in K}|f(s^{-1}r)-f(r)|\longrightarrow0.

For general $f\in L^2(G)$ approximate by $f_0\in C_c(G)$ and use

\|L_sf-f\|_2\le2\|f-f_0\|_2+\|L_sf_0-f_0\|_2.

This proves continuity at the identity, and the representation law proves it everywhere. Compact-neighborhood estimates avoid any invalid dominated-convergence argument for arbitrary nets. $\square$

On the tensor space $\mathcal H=H\otimes L^2(G)$ put

N=M\,\overline\otimes\,D,\qquad V_s=U_s\otimes L_s,\qquad \beta_s=\alpha_s\otimes\ell_s.

The representation $V$ is strongly continuous. On elementary tensors this follows from the triangle inequality and the strong continuity of both factors, and density plus the norm-one bound extends it to $\mathcal H$ . Multiplication operators satisfy $L_sm_hL_s^*=m_{\ell_sh}$ , so $V_s$ implements $\beta_s$ on the generators of $N$ , hence on all of $N$ by normality.

OA-FLOW.AWC.UNTWIST — Constructing a unitary without a field decomposition

We will use the formula

(W\xi)(r)=U_r\xi(r),

but construct it in $H\otimes L^2(G)$ before making any assertion about arbitrary sections.

A norm-continuous compactly supported function $\eta:G\to H$ gives a vector in $H\otimes L^2(G)$ . To check this without separability of $H$ , its range on its compact support is a compact subset of a metric space, and thus has a finite $\varepsilon$ -net. Let $P$ be the orthogonal projection onto the finite-dimensional span of that net. Then $\|\eta(r)-P\eta(r)\|\le\varepsilon$ on the support, while $P\eta$ is a finite sum of fixed Hilbert vectors multiplied by scalar $C_c(G)$ functions. The resulting $L^2$ error is at most $\varepsilon$ times the square root of the finite measure of the support. This identifies $\eta$ with a limit of elementary tensor vectors and gives its norm by integration. Finite sums of scalar $C_c$ functions times Hilbert vectors are dense in the tensor space by construction.

For such a compactly supported continuous $\xi$ , both $r\mapsto U_r\xi(r)$ and $r\mapsto U_r^*\xi(r)$ are again compactly supported and norm continuous. The vector estimate in OA-FLOW.AWC.VECTORCONT proves the continuity, and pointwise unitarity gives

\int_G\|U_r\xi(r)\|^2\,dr=\int_G\|\xi(r)\|^2\,dr.

Thus the two pointwise operations extend to isometries of $\mathcal H$ . Their compositions are the identity on the dense compactly supported continuous class, so they are mutually inverse on $\mathcal H$ . This constructs a unitary $W$ with the claimed formula and adjoint formula $W^*\xi(r)=U_r^*\xi(r)$ . No range of a general global section was declared separable, and no direct-integral disintegration was used.

Proposition. Conjugation by $W^*$ restricts to a normal automorphism $\gamma$ of $N$ , and

\gamma\beta_s\gamma^{-1}=\operatorname{id}_M\otimes\ell_s.

Proof. First $W$ commutes with scalar continuous multiplication operators by its defining formula. It therefore commutes with $1\otimes D$ , the von Neumann algebra they generate.

For $a\in M$ , let $A_a=W^*(a\otimes1)W$ . On the dense continuous test vectors,

(A_a\xi)(r)=U_r^*aU_r\xi(r)=\alpha_{r^{-1}}(a)\xi(r).

For $b'\in M'$ , the operator $\alpha_{r^{-1}}(a)$ commutes with $b'$ . The formula therefore proves that $A_a$ commutes with $b'\otimes1$ . It also commutes with $1\otimes D$ , since all three factors in its definition do. Hence it belongs to $(M'\overline\otimes D)'=N$ by the tensor commutant contract. The same argument for $W(a\otimes1)W^*$ uses the coefficient $\alpha_r(a)$ and again gives membership in $N$ .

Since $M\otimes1$ and $1\otimes D$ generate $N$ , these conclusions and normality of spatial conjugation give both $W^*NW\subset N$ and $WNW^*\subset N$ . Conjugating the second inclusion by $W^*$ supplies the reverse of the first. Thus $W^*NW=N$ , and $\gamma=\operatorname{Ad}(W^*)|_N$ is a normal automorphism.

Finally compute the conjugated implementing unitary on a continuous test vector:

\begin{aligned} (W^*V_sW\xi)(r) &=U_r^*U_sU_{s^{-1}r}\xi(s^{-1}r)\\ &=\xi(s^{-1}r). \end{aligned}

The representation law $U_sU_{s^{-1}r}=U_r$ is valid for nonabelian $G$ with precisely this order. Density gives $W^*V_sW=1\otimes L_s$ on $\mathcal H$ . Conjugating $N$ by these unitaries proves the asserted action identity. $\square$

This proof establishes membership in the spatial tensor algebra by commutants. It does not assume that every bounded operator-valued measurable field has a tensor-product realization at arbitrary nonseparable generality.

OA-FLOW.AWC.INVARIANT — An auxiliary invariant weight

Let

\psi=\varphi\otimes\mu\quad\hbox{on }N,\qquad \chi=\psi\circ\gamma.

Both are n.s.f. by the tensor-weight and automorphism-pullback contracts. The weight $\psi$ is invariant under $\operatorname{id}\otimes\ell_s$ , since tensor naturality and left invariance of Haar integration give

\psi\circ(\operatorname{id}\otimes\ell_s)=\varphi\otimes(\mu\circ\ell_s)=\varphi\otimes\mu.

The preceding proposition then yields

\chi\circ\beta_s=\psi\circ\gamma\circ\beta_s =\psi\circ(\operatorname{id}\otimes\ell_s)\circ\gamma=\chi.

The invariant weight is on $N$ , not necessarily on $M$ . This construction does not average $\varphi$ over $G$ , does not normalize Haar measure to a probability measure, and does not require the original weight to be invariant.

OA-FLOW.AWC.IDENTITY — Expressing every derivative using one curve

Define the single continuous unitary curve

v_t=[D\psi:D\chi]_t\in\mathcal U(N).

The derivative chain rule gives $[D\chi:D\psi]_t=v_t^*$ . For each $s$ , use invariance of $\chi$ , naturality, and then the chain rule to obtain

\begin{aligned} [D(\psi\circ\beta_s):D\psi]_t &=[D(\psi\circ\beta_s):D(\chi\circ\beta_s)]_t\,[D\chi:D\psi]_t\\ &=\beta_s^{-1}(v_t)v_t^*. \end{aligned}

Every weight in the first line is on $N$ . On the other hand, the tensor contracts give

\begin{aligned} [D(\psi\circ\beta_s):D\psi]_t &=[D((\varphi\circ\alpha_s)\otimes(\mu\circ\ell_s)):D(\varphi\otimes\mu)]_t\\ &=[D(\varphi\circ\alpha_s):D\varphi]_t\otimes[D(\mu\circ\ell_s):D\mu]_t\\ &=c(s,t)\otimes1. \end{aligned}

Therefore

c(s,t)\otimes1=\beta_s^{-1}(v_t)v_t^*=V_s^*v_tV_s v_t^*.\tag{AWC}

This is the central calculation. The scalar Haar algebra carries $\ell_s$ , while $M$ carries $\alpha_s$ . A composition $\mu\circ\alpha_s$ would have the wrong domain. The last tensor factor is $1$ because $\mu\circ\ell_s=\mu$ , not because $G$ is unimodular. Finally, $v_t$ need not be a unitary representation of $\mathbb R$ ; its strong continuity and unitarity are all that the next step uses.

OA-FLOW.DW.COCYCLECONT — Joint continuity at full group generality

Theorem, relative to the exact imports above. The map

G\times\mathbb R\longrightarrow\mathcal U(M),\qquad (s,t)\longmapsto[D(\varphi\circ\alpha_s):D\varphi]_t

is jointly $\sigma$ -strong-* continuous. In particular it is jointly $\sigma$ -strongly continuous. All hypotheses are those of OA-FLOW.AWC.SETTING; none of the excluded countability, invariance, or unimodularity conditions is added.

Proof. The curve $v_t$ is $\sigma$ -strongly continuous on $N$ by the weight-derivative contract. In the faithful normal tensor representation every vector functional is normal, so this implies strong continuity on $\mathcal H$ . Its adjoint is also strongly continuous by the unitary inversion estimate. The representation $s\mapsto V_s$ was proved strongly continuous above. Equation (AWC) and OA-FLOW.AWC.VECTORCONT show that $(s,t)\mapsto c(s,t)\otimes1$ is jointly strong-* continuous on $\mathcal H$ .

To recover exactly the claimed topology on $M$ , choose any unit vector $\zeta\in L^2(G)$ . Such a vector exists because a nonzero compactly supported continuous function has finite positive $L^2$ norm after choosing it nonzero on a nonempty open set. For $\omega\in M_*^+$ , use its standard-form vector $\xi_\omega\in H$ . If $(s_i,t_i)\to(s,t)$ is any net and $c_i=c(s_i,t_i)$ , $c=c(s,t)$ , then

\begin{aligned} p_\omega(c_i-c) &=\|(c_i-c)\xi_\omega\|\\ &=\|((c_i-c)\otimes1)(\xi_\omega\otimes\zeta)\|\longrightarrow0. \end{aligned}

The same equality with $c_i^*-c^*$ , using the strong continuity already proved for the adjoint of the tensor expression, gives $p_\omega(c_i^*-c^*)\to0$ . These are all the defining seminorms, which proves joint $\sigma$ -strong-* continuity. $\square$

The proof does not evaluate an arbitrary $L^\infty$ representative at the identity of $G$ . The coefficient is recovered through a genuine tensor embedding and its vector seminorms. It also does not infer joint continuity from the separate continuity of the two parameters.

OA-FLOW.AWC.LAWS — Two parameters, two cocycle identities

The resulting function has useful algebraic checks in both variables. They also determine the order of the factors in formulas used later for dual weights.

Proposition. For $s,r\in G$ and $t,u\in\mathbb R$ ,

c(s,t+u)=c(s,t)\sigma_t^\varphi(c(s,u)),\tag{T}

c(sr,t)=\alpha_r^{-1}(c(s,t))c(r,t),\tag{G}

c(e,t)=1,\quad c(s,0)=1,\quad c(s^{-1},t)^*=\alpha_s(c(s,t)).\tag{I}

For a fixed $t$ , the function

d_t(s)=\alpha_s(c(s,t))=c(s^{-1},t)^*

is an ordinary $\alpha$ -cocycle:

d_t(sr)=d_t(s)\alpha_s(d_t(r)).

It satisfies

\sigma_t^\varphi\alpha_s\sigma_{-t}^\varphi =\operatorname{Ad}(d_t(s))\alpha_s.

Proof. Equation (T) is the derivative cocycle law with denominator $\varphi$ . For (G), remember that $\varphi\circ\alpha_{sr}=(\varphi\circ\alpha_s)\circ\alpha_r$ . Insert $\varphi\circ\alpha_r$ as the middle weight in the chain rule and then use naturality:

\begin{aligned} c(sr,t) &=[D((\varphi\circ\alpha_s)\circ\alpha_r):D(\varphi\circ\alpha_r)]_t\, [D(\varphi\circ\alpha_r):D\varphi]_t\\ &=\alpha_r^{-1}(c(s,t))c(r,t). \end{aligned}

The first two normalizations in (I) follow from equal-weight and zero-time normalization. Substituting $r=s^{-1}$ in (G) gives $1=\alpha_s(c(s,t))c(s^{-1},t)$ , proving the last identity. Applying $\alpha_{sr}$ to (G), in the stated order, gives

d_t(sr)=\alpha_s(c(s,t))\,\alpha_s(\alpha_r(c(r,t)))=d_t(s)\alpha_s(d_t(r)).

Finally naturality and modular implementation say

\alpha_s^{-1}\sigma_t^\varphi\alpha_s =\operatorname{Ad}(c(s,t))\sigma_t^\varphi.

Multiplying on the left by $\alpha_s$ and on the right by $\sigma_{-t}^\varphi$ yields the claimed conjugated action. $\square$

Joint continuity of $(s,t)\mapsto d_t(s)$ follows directly from $d_t(s)=c(s^{-1},t)^*$ and the theorem. Equation (G) is a pullback identity; replacing it by a left cocycle law for $c$ itself would change both the order and the action. The transformed function $d$ is the left cocycle.

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