Introduction
Measure equivalence and non-commutativity
Suppose that π : Λ → U(H) and ρ : Λ → U(K) are unitary representations and Λyσ(M,Tr) is a trace-preserving action on a semifinite von Neumann algebra. Suppose that π : Λ → U(H) is a mixing representation and Λyσ(M,Tr) is a trace-preserving action on a semifinite von Neumann algebra.
Preliminaries
Von Neumann algebras: Definition and Examples
- The standard representation
- Group von Neumann algebras
- Group Measure Space Construction
A von Neumann algebra M is called trace if it admits a linear functional τ : M →C, called atrace, which is. The group gauge space von Neumann algebraL∞(X)oΓ ⊂ B(H) is defined as the WOT closure of the linear span of{f ug | f ∈ L∞(X), g ∈Γ}.
Semi-finite traces
If Mi is a semifinite von Neumann algebra with a faithful normal semifinite trace Tr, and p ∈ M is a projection of finite traces, then the map x 7 → Tr(xp) is a weak operator topology continuous. Let Mis be a semi-finite von Neumann algebra with a faithful normal semi-finite traceTr,x ∈ Mis compact, and {pi} be a net of projections from finite traces, such that pi → 0 in the weak operator topology.
Actions on semi-finite von Neumann algebras
If we have two embeddings θ1, θ2 : B(H) → M, then θ1(B(H)) andθ2(B(H)) are conjugate with a unit inMif and only if for some series one projection p ∈ B(H) we have that θ1(p) and θ2(p) are Murray-von Neumann equivalent2. Note that by Fell's absorption principle3, the representationΓ3γ 7→uγ ∈ MoΓ is conjugate to a multiple of the left regular representation and therefore generates a copy of the group von Neumann algebraLΓ.
The basic construction
Tensor products of operator spaces
The second is the Hilbert rice spaceHr, which assigns H with the operator space structure that comes from the canonical isomorphism H ∼= CB(H,C). The operator space structure onE ⊗min F is independent of the concrete representations, and we have a completely isometric embedding.
Hilbert C ∗ -modules
Every ultraweakly closed subspace E of B(H) is a dual operator space with a canonical predual B(H)∗/E⊥, where E⊥ is the preannihilator of E. We note that even in the case where F = Mis a von Neumann algebra this embedding will in general not be surjective.
Measurable functions into separable Banach spaces
Another corollary we will use is that ifKandHare are two double HilbertM modules,X ⊂ K is a weakly∗-dense M-invariant subset, andV : If E = (E∗)∗ is a dual Banach space and K ⊂ E is a weakly∗-closed convex subset, then we have isomorphism below the above.
Properly proximal groups
We now claim that the set of proper proximal points in Prob(X) is closed with convex combinations. A group Γ is defined in [BIP18] to be properly proximal if there is an action of Γ on a compact Hausdorff space X such that there is no Γ-invariant measure on X and such that Prob(X) have a proper proximal end. Let be an infinite discrete group. ii) Γ has a proper proximal action on a compact Hausdorff space that does not have an in-variant measure. iii).
There is a dual BanachΓ-moduleE and a non-emptyΓ-invariant weak∗-compact convex subsetK ⊂E such that K has a proper proximal point (with respect to the weak∗ topology) but has no fixed point. iv). There is a dual Banach Γ-module E and a nonempty Γ-invariant weak∗-compact convex subsetK ⊂ E such that the actionΓyK is properly proximal (with respect to the weak∗ topology) but has no fixed point. v) Γhas an action of affine homeomorphisms on a nonempty compact convex subset K of a locally convex topological vector space such that the actionΓyK is properly proximal but. i) =⇒ (iii) is trivial by considering the weakly∗-compact convex set of probability measures on a compact Hausdorff space. i) =⇒ (ii) is also trivial since we can restrict to the closure of a path in a proper proximal point. ThenΓ is properly proximal if and only if there is a dual BanachΓ-moduleE such that the induced mapHb1(Γ, Emix) → Hb1(Γ, E) has a non-trivial range.
Let Γ be correctly proximal and letΓyX be an action on a compact Hausdorff space such that Prob(X) has a correct proximal point η, but X has no invariant measure. If we let K be the weak∗-closure of c(Γ), then K is weak∗-compact by the Banach-Alaoglu Theorem, and we know that ΓyαK is correctly proximal.
Multipliers on discrete groups and associated multiplier algebras
The index of the von Neumann coupling is the ratio Tr(p)/Tr(q) and is denoted by [Γ : Λ]M. Then N ⊗ M has a Λ-fundamental domain, and the induced semi-finite trace on (N ⊗ M)Λ gives a (Σ,Γ)von Neumann coupling with index. Assume that Λyσ(M,Tr) is a trace-preserving action on a semi-finite von Neumann algebra, and π : Λ → U(H) is a unitary representation.
Suppose that π : Λ → U(H) is a unitary representation and Γ×Λy(M,Tr) is a trace-preserving action on a semifinite von Neumann algebra such that the action Λ admits a fundamental domain of finite trace. Suppose that π : Λ → U(H) is a unitary representation and Λyσ(M,Tr) is a trace-preserving action on a semifinite von Neumann algebra having a fundamental domain of finite traces p. Let M be a semifinite von Neumann algebra with faithful normal semifinite traceTrandΛyσ(M,Tr) a trace-preserving action with fundamental domain of finite trace p.
Similarly, we have a fundamental domain with a finite trace for the action of ΛonM, and therefore we see that this is then a Γ-Λvon Neumann coupling. We similarly have a fundamental domain with finite traces for LΛinN, and therefore N is an LΓ-LΛvon Neumann coupling.
Von Neumann equivalence
Fundamental domains for actions on von Neumann algebras
If p∈ Mis is a fundamental domain, one can work equivalently with the Γ-equivariant normal inclusion θp: `∞Γ → M given by θp(f) = P. Suppose M is a semifinite von Neumann algebra with a semi-finite normal traceTr, and Γyσ(M,Tr) is a trace-preserving action with a fundamental domain p. i) The mapτ(x) = Tr(pxp)is independent of the fundamental domain and defines a faithful normal semi-finite trace onMΓ. Here MΓ denotes the space of all Γ-fixed points. is a strong operator topology with density in M. vi).
Since Tr is semi-finite, there exists an increasing net of projections of finite traces {qi}i∈I such that qi → p in the weak operator topology. Note that since it is a fundamental domain, this sum converges in the strong operator topology, and we have aTγ ∈ MΓ. If α ∈Aut(M) is a Γ-equivariant automorphism that preserves Tr, and ifpis is a Γ-fundamental domain, then α(p) is also a Γ-fundamental domain, and so forx∈ MΓ holds.
Von Neumann couplings
As in the case of measure equivalence, von Neumann equivalence is an equivalence relation. Both (i) and (ii) in Proposition 4.15 show that the Haagerup property is preserved under von Neumann equivalence. Let M be a semifinite von Neumann algebra with faithful normal semifinite trace Trand let Λyσ(M,Tr) be a trace-preserving action with fundamental domain of finite tracep.
The fundamental domain M within M consists of a realization of the standard representation M ⊂ B(L2(M)) as an intermediate von Neumann subalgebraM ⊂ B(L2(M))⊂ M. LetM ⊂ Mbe an inclusion of von Neumann algebras with Mbeing-end and semi M being finite and σ-finite. We then define the fusion (or composition) of MMN and NMQ unions to form von Neumann algebras.
Von Neumann equivalence gives an equivalence relation on the set of finite, σ-finite von Neumann algebras. AΓ-fundamental domain in M gives aΓ-equivariant embedding`∞Γ⊂ And so we get an embedding of von Neumann algebras.
Von Neumann equivalence and group approximation properties
Inducing actions via semi-finite von Neumann algebras
- Properly proximal actions
Note that if X = (X∗)∗ is a dual Banach Γ-module, then we can also regard X as an operator Γ-module by giving X∗ with the operator space structure min(X∗), so that max( X) ) = (min(X∗))∗ becomes a dual operator Γ-module. Therefore, we have an action Γy(M ⊗K)Λ, which we refer to as theΓ-action induced from theΛ-actionΛyK. Since our weak operator topology has convergenceeuαn(A)→ 0, and since τ is a finite trace on MΛ, it follows that for any finite setF ⊂Λ we have.
Sincek is properly proximal, and since {αn(A)} is uniformly bounded in the trace norm, it follows that there is weak∗-convergence. If we take linear combinations of vectors of the formλt⊗v, it follows that for allz ∈CΛ⊗algMΛ we have a weak∗-convergence. If this is the case, we find that from Lemma 4.6, because traces are preserved.
If the action ΛyK has a point that is properly proximal, then so does the induced actionΓy(M ⊗K)Λ. If A ∈ mTr, then if the action of Γis mixing, we have thatσγn(A) converges to0in the weak operator topology.
Inducing unitary representations
From Theorem 3.3, the existence of a fundamental domain for Γ implies that the Koopman representation is a multiple of the left-regular representation, and thus is mixing for any infinite group. If we also get a track-preserving action Γyσ(M,Tr) that commutes together with the Λ-action, then we see that (4.2) also holds for the Γ-action. As described in Section 2.6, it follows that Vp has a weak∗-continuous extensionVp :MΛ⊗ H → (M ⊗ H)Λ that preserves the inner product; and to see that V is surjective, it suffices to show that the range of Vp is dense when viewed as a map in (M ⊗ H)Λτ, where τ is the trace given by Proposition 3.3 , i.e. τ(x) = Tr( pxp)forx∈ MΛ.
In the case where Mis is related to the equivalence aW∗ as in Proposition 4.12, this follows from the results in [Cho83, CJ85]. Suppose π : Λ →U(H)andδρ: Λ →U(K)are two unitary representations of Λ, andΓ×Λy(M,Tr)is a von Neumann union. ii) If π is mixing, then πM is mixing. iii) λMis a multiple of the left-regular representation ofΓ. iv) If π is weakly mixed, then πM has no nonzero invariant vectors. Suitability is characterized by the fact that the left regular representation weakly contains the trivial representation, so (i) and (iii) in Proposition 4.15 show that suitability is preserved under von Neumann equivalence.
Finally, if Γ has property (T) and π is a representation of Λ that weakly contains the trivial representation, then since 1M contains the trivial representation for Γ, it follows that πM also weakly contains the trivial representation. Property (T) then implies that πM contains non-zero Γ-invariant vectors, and from (iv) in Proposition 4.15 it follows that π is not weakly mixed.
Inducing Herz-Schur Multipliers
The notion of von Neumann equivalence also allows a generalization in the settings of finite von Neumann algebras, which we study in this chapter. Two finite, σ-finite von Neumann algebras M and N are Neumann equivalent, denoted by M ∼vN E N, if there exists a von Neumann coupling between them. Reflexivity follows by considering the trivial von Neumann coupling B(L2(M, τ)) with standard embeddings M and Mop.
We can now show the relationship between the von Neumann equivalence for groups and for finite von Neumann algebras as stated in Theorem 1.6. It would be interesting to have examples of von Neumann algebras, or even Γ-groups, like this. We also know of no examples of groups that are von Neumann equivalent but not measure equivalent.
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Von Neumann equivalence for finite von Neumann algebras
