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

Changing the Hilbert space of a regular crossed product

CC0 1.0.

The continuity inputs are proved in Continuity of actions. The Haar and Radon foundations and spatial tensor comparison supply the other indicated inputs.

A regular crossed product is represented on a space of Hilbert-valued functions on the group. Changing the Hilbert space changes that concrete representation. The aim is to prove that its named coefficient and group operators still determine the same von Neumann algebra, through a unique normal isomorphism. Scalar matrix coefficients make this comparison possible even when the two Hilbert spaces have different dimensions.

OA-FLOW.REG.SETTING — The two representations being compared

Let $G$ be an arbitrary locally compact Hausdorff group, with fixed left Haar measure $\mu$, and let $M$ be a von Neumann algebra. Let $\alpha:G\to\operatorname{Aut}(M)$ be point-ultraweakly continuous. By OA-FLOW.TOP.PREDUAL and OA-FLOW.TOP.JOINT, its predual orbits are norm continuous, and its evaluation on bounded algebra sets is jointly sigma-strong-star continuous.

For a normal unital representation $\rho:M\to B(H_\rho)$, use

Inner products are linear in the first variable. Sections are strongly measurable, square-integrable functions modulo null sets in the localizable Haar completion. Neither the group nor either Hilbert space is assumed countable, separable, sigma compact or unimodular. There is no weight in this construction.

We first allow $\rho$ to have a kernel. The comparison theorem will use faithful normal unital representations $\rho$ and $\sigma$ of the same algebra. If $M=0$, unitality forces their Hilbert spaces to be zero and the conclusions have their unique zero interpretation. We therefore carry out the nonzero case.

OA-FLOW.REG.IMPORT.FOUNDATIONS — Exact background inputs

The following mathematical foundations are used with their stated hypotheses.

• Haar measure is regular, finite on compact sets and positive on nonempty open sets. Compact-support continuous cutoffs exist inside identity neighborhoods. Scalar $C_c(G)$ is dense in $L^1(G)$ and $L^2(G)$. The Hilbert-space section identification (R1), density of finite sums $x\otimes h$ with $h\in C_c(G)$, scalar integral inequalities and left Haar invariance are included. The displayed group integrals below are all in one Haar variable; no interchange of integrations on a general product space is used.
• For a von Neumann algebra, $M=(M_*)^*$, normal bounded linear maps have bounded preadjoints, and positive maps are normal exactly when they preserve suprema of bounded increasing positive nets. Normal automorphisms preserve these suprema. Normal unital representations are contractive and have von Neumann images; faithful ones are isometric. A faithful normal unital representation identifies $M$ normally with its concrete image; its inverse on that image is normal.
• We use elementary Hilbert tensor products, adjoints and bounded sesquilinear forms, the bicommutant theorem, and the spatial tensor product as the von Neumann algebra generated by elementary tensors. Bounded increasing positive nets in a concrete von Neumann algebra converge strongly to their supremum. Bounded strong convergence implies ultraweak convergence, and normal functionals admit the absolutely summable vector-functional expansion stated in OA-FLOW.TOP.FOUNDATIONS.

The one infinite tensor-extension result used here is already proved in part (d) of OA-FLOW-DUAL-MULTIPLICITY: a normal isomorphism $A\to B$ extends uniquely and normally to $A\overline\otimes B(K)\to B\overline\otimes B(K)$, with its stated action on elementary tensors. That finite-corner proof has arbitrary Hilbert spaces as its hypotheses. It uses the background operator facts just listed, not the regular-crossed-product or Fourier conclusions elsewhere in lesson 03. We use exactly that proved part and its matrix-coefficient formula, so this lesson does not assume the representation independence it is about to establish.

OA-FLOW.REG.CONSTRUCTION — Bounded coefficient operators and left translations

For $\xi\in C_c(G,H_\rho)$, set

Proposition. These formulas extend to a unital star representation $\pi_\rho:M\to B(\mathcal H_\rho)$ and a strongly continuous unitary representation $t\mapsto L^\rho_t$. They satisfy

For every fixed $a,t,\xi$, formula (R2) holds almost everywhere for an arbitrary section $\xi\in\mathcal H_\rho$.

Proof. The operator orbit $s\mapsto\rho(\alpha_{s^{-1}}(a))$ is strongly-star continuous. To see that the possibly nonfaithful $\rho$ causes no problem, apply the intrinsic sigma-strong-star statement from lesson 11 to the normal positive functionals $x\mapsto\langle\rho(x)v,v\rangle$. Their quadratic forms give the required squared vector norms in $H_\rho$. The orbit is bounded by $\|a\|$.

If $\xi$ varies with $s$, the estimate

proves continuity. The output has compact support contained in that of $\xi$. Its norm estimate integrates to

Density therefore gives a bounded extension. On continuous compact-support vectors, multiplication, linearity and the identity follow pointwise from the same properties of $\rho$ and $\alpha_s$. For the adjoint, integrating the pointwise inner-product identity gives

Density extends these identities to all vectors, proving that $\pi_\rho$ is a unital star representation.

Left Haar invariance gives $\|L^\rho_t\xi\|_2=\|\xi\|_2$. The pointwise group law and inverse $L^\rho_{t^{-1}}$ show that the extensions are unitary. To prove strong continuity at the identity, take $\xi\in C_c(G,H_\rho)$, with support $K$, and choose a compact identity neighborhood $V$. For $t$ near the identity, the difference $s\mapsto\xi(t^{-1}s)-\xi(s)$ is supported in $VK\cup K$, a fixed compact set. Joint continuity and a finite subcover give

The support has finite measure, so the difference tends to zero in $L^2$. Approximation by these vectors and the unitary norm bound extend this convergence to every $\xi$. The group law proves continuity at every $t$.

On the same dense section space,

This proves covariance. No modular factor occurs: all translations used here are left translations for left Haar measure.

For a general section, choose continuous compact-support approximants in $L^2$. A subsequence converges almost everywhere, both before and after applying a fixed bounded operator in (R2). Indeed, choose the subsequence so that the sum of the squared $L^2$ errors of inputs and outputs is finite. Scalar monotone convergence makes the sum of the pointwise squared errors finite almost everywhere, which gives both pointwise limits. The bounded pointwise coefficient operators identify those limits; left translation preserves null sets and gives the same conclusion for $L^\rho_t$. Thus the output field has a strongly measurable representative. This is a statement for a fixed operator and section; it requires no universal exceptional null set. $\square$

OA-FLOW.REG.NORMALITY — Compact monotonicity handles arbitrary nets

Theorem. The representation $\pi_\rho$ is normal. Its kernel is

In particular, if $\rho$ is faithful then $\pi_\rho$ is faithful and isometric.

Proof. Let $0\leq a_i\uparrow a$ be a bounded increasing net in $M$. Fix $\xi\in C_c(G,H_\rho)$, with compact support $K$. For each index $i$, the scalar function

is continuous and nonnegative on $K$. It decreases with $i$. At every fixed $s$, normality of $\alpha_{s^{-1}}$ and $\rho$ implies $F_i(s)\downarrow0$.

For $\varepsilon>0$, the open subsets $\{s\in K:F_i(s)<\varepsilon\}$ cover $K$. Choose a finite subcover and then a single index dominating its finitely many indices. Since the functions decrease, every larger index has $F_i<\varepsilon$ throughout $K$. Consequently $\sup_K F_i\to0$, and

This is a compactness argument for a directed net, not an interchange of a measure integral with an arbitrary monotone net.

The positive operators $\pi_\rho(a-a_i)$ are uniformly bounded by $\|a\|$. If $\eta\in\mathcal H_\rho$ and $\xi$ approximates it in $L^2$, the difference of their quadratic forms has absolute value at most

Thus (R6) holds for every $\eta$. The bounded increasing net $\pi_\rho(a_i)$ has a supremum, and its quadratic forms show that this supremum is $\pi_\rho(a)$. The normality criterion in the foundations proves normality.

If all the pointwise coefficients in (R4) vanish, (R2) gives $\pi_\rho(a)=0$. Conversely, suppose $\rho(\alpha_{s_0^{-1}}(a))v\ne0$ for some $s_0,v$. Continuity gives an open neighborhood $U$ of $s_0$ and $c>0$ on which the norm of this vector is at least $c$. Choose a nonzero scalar $h\in C_c(G)$ supported in $U$. Haar positivity gives $\|h\|_2>0$, and

Thus $\pi_\rho(a)\ne0$. This proves the first equality in (R4); applying $\alpha_s$ gives the second. If $\rho$ is faithful, evaluating the pointwise condition at the identity forces $a=0$. A faithful star representation is isometric by the foundational $C^*$-norm fact. $\square$

Faithfulness is tested on an open neighborhood, not by assigning positive measure to the single identity element. For a nondiscrete group that singleton can have measure zero.

OA-FLOW.REG.SUPPORT — If a representation is allowed to be degenerate

The unital convention in the comparison theorem is necessary. More generally, suppose $\rho:M\to B(H_\rho)$ is normal but is not required to be unital. Put

The projection $p$ is the identity on every value of $\rho$. Its compression $\rho_+:M\to B(H_+)$ is normal and unital; it is faithful when $\rho$ is faithful. The formulas (R2) still define bounded coefficient operators and unitary translations. All the preceding construction, normality and kernel arguments apply, with $\pi_\rho(1)=p\otimes1$ in place of the identity.

Under the orthogonal decomposition

the coefficient operators are $\pi_{\rho_+}(a)\oplus0$, and the group operators are $L^+_t\oplus L^-_t$. The projection $P=p\otimes1=\pi_\rho(1)$ belongs to their generated von Neumann algebra and commutes with every generator. It is therefore central in that algebra. Consequently

To verify the formula, multiply each generator by $P$ and $1-P$. In the first corner the resulting generators are exactly the compressed regular pair. In the second corner all coefficients vanish and the remaining generators are the scalar left regular representation with multiplicity $H_-$. The central projection separates the two corners, so these identifications give both inclusions. Omit a summand when its Hilbert space is zero.

Thus the faithful-model comparison proved below always applies on the coefficient-support subspaces. On the uncompressed spaces, a degenerate model can have the extra group-algebra summand. For example, let $G$ be the one-element group and compare the faithful normal maps

The generated algebras are $M_2(\mathbb C)$ and $M_2(\mathbb C)\oplus\mathbb C$. In particular, a generator-preserving isomorphism on the full spaces would have to take $\pi_\rho(1)=1$ to the proper projection $\pi_\sigma(1)$, which is impossible for an algebra isomorphism. This explains exactly why “faithful normal representation” must either include nondegeneracy by convention or be followed by compression to its support.

OA-FLOW.REG.AVERAGES — Ultraweak $L^1$ averages and matrix coefficients

For $h\in L^1(G)$ and $a\in M$, define $A_h(a)\in M$ by

Lemma. Formula (R7) defines a unique element with

For every normal bounded linear map $\Phi:M\to N$,

The integral on the right has the same scalar-pairing meaning. No operator-norm Bochner integral is asserted.

Proof. The scalar orbit in (R7) is continuous and bounded by $\|\omega\|\|a\|$. Its product with $h$ is integrable. The integral is a bounded linear functional of $\omega$, with norm at most $\|h\|_1\|a\|$. Since $(M_*)^*=M$, it defines the stated element uniquely. If $\nu\in N_*$, normality of $\Phi$ gives $\nu\circ\Phi\in M_*$. Substitute that functional in (R7). The resulting scalar identity is (R8), because normal functionals separate elements of $N$. $\square$

For $u\in L^2(G)$, let $V^\rho_u:H_\rho\to\mathcal H_\rho$ be the bounded map $V^\rho_u x=x\otimes u$. The coefficient operators of the regular copy are

Indeed $\overline u v\in L^1(G)$ by Cauchy–Schwarz. For $x,y\in H_\rho$, the scalar pairing of the left side is

This follows first for continuous compact-support $u,v$ from (R2), and then for all $u,v$ by $L^2$ approximation and the $L^1$ bound on their products. Apply (R7) to the normal functional $b\mapsto\langle\rho(b)x,y\rangle$. Since these vectors are arbitrary, (R9) follows. The scalar conjugation in $\overline u v$ reflects our first-variable-linear convention.

OA-FLOW.REG.TENSORLOCATION — Locating the regular algebra in one tensor product

Write $K=L^2(G)$. Define the concrete regular crossed product

Proposition. For every normal unital $\rho$,

Proof. Here is the finite-corner test that we need. If $T\in B(H\otimes K)$ has all its coefficient operators $V_u^*TV_v$ in a concrete von Neumann algebra $A\subset B(H)$, choose any orthonormal basis $(e_i)_{i\in I}$ of $K$. There is no countability requirement on $I$. For finite $J\subset I$, set $p_J=\sum_{i\in J}E_{ii}$. Then

As $J$ increases over finite subsets, the projections converge strongly to $1$, so the compressions converge strongly to $T$. Strong closedness of the spatial tensor product proves $T\in A\overline\otimes B(K)$. This is the finite-corner argument also used in part (b) of OA-FLOW-DUAL-MULTIPLICITY.

Apply this test to (R9), with $A=\rho(M)$. Finally $L^\rho_t=1_{H_\rho}\otimes L_t$ belongs to the same tensor product. That tensor product is a von Neumann algebra containing all the generators of (R10), so it contains their generated algebra. $\square$

The integral in (R7) takes values in the abstract coefficient algebra before a representation is selected. Equation (R9) is therefore a way to compare regular operators without comparing their Hilbert vectors point by point.

OA-FLOW.REG.INDEPENDENCE — The normal comparison and its generators

Theorem. Let $\rho:M\to B(H_\rho)$ and $\sigma:M\to B(H_\sigma)$ be faithful normal unital representations. There is a unique normal isomorphism

such that

For a third faithful normal unital representation $\tau$, these maps satisfy

In particular $C_{\rho,\sigma}^\alpha$ is the inverse of $C_{\sigma,\rho}^\alpha$.

Proof. The map

is a normal isomorphism of concrete von Neumann algebras. By the proved finite-corner tensor-extension theorem, it has a unique normal extension

That proof identifies every matrix coefficient of $\Theta(T)$ with $\gamma$ applied to the corresponding coefficient of $T$. This identity also holds for coefficient pairs $u,v\in K$: one may first choose an orthonormal basis of their finite-dimensional span and extend it to a basis of $K$, and then use linearity in the two coordinates. Equations (R9) and (R15) give

Thus $\Theta(\pi_\rho(a))=\pi_\sigma(a)$, since elementary tensor vectors span a dense space and determine a bounded operator. Equivalently, the middle step transports the ultraweak integral through the normal map $\gamma$, as justified by its preadjoint in (R8). No spatial unitary implementing $\gamma$ has been chosen.

The tensor extension takes $1\otimes L_t$ to $1\otimes L_t$. It therefore takes each group generator to its counterpart. Since $\Theta$ and its inverse are normal, they carry the ultraweakly closed algebras generated by the two named families onto each other. Its restriction gives the desired normal isomorphism.

Any two normal star homomorphisms with (R13) agree on the unital star algebra of finite words in the named generators. That algebra is ultraweakly dense in the generated von Neumann algebra, so normality gives equality everywhere. The composites in (R14) have the same generator values; this uniqueness proves both identities and the inverse assertion. $\square$

The result is a normal algebra isomorphism between concrete models. It need not be implemented by a unitary between the particular Hilbert spaces originally chosen.

OA-FLOW.REG.SYSTEMNATURALITY — Relabeling the coefficient system

Suppose $(P,\beta)$ is another von Neumann system over the same $G$, and $\eta:M\to P$ is a normal isomorphism satisfying

Choose faithful normal unital $\rho$ of $M$ and $\sigma$ of $P$. There is a unique normal isomorphism

Proof. Use the normal coefficient isomorphism $\sigma\eta\rho^{-1}$ in (R15). Equivariance and (R8) give

Consequently its matrix coefficients satisfy the analogue of (R16). The proof of restriction and uniqueness is unchanged. Composition of coefficient-system isomorphisms agrees with composition of the resulting crossed-product maps, because their values on every named generator agree. $\square$

This functorial statement concerns normal isomorphisms. It is not a universal property for arbitrary covariant representations or a statement about arbitrary homomorphisms.

Editable source · Proof dependencies and component terms