The purpose of this dissertation is to study the notion of proper proximity in the context of groups and von Neumann algebras, as well as various stiffness phenomena of proper proximal groups and von Neumann algebras. While the stiffness properties of proximal von Neumann algebras were demonstrated in Chapter 2, the purpose of this chapter is to show that proper non-proximal groups also possess some stiffness phenomena.
Background on von Neumann algebras
We generalize the notion of proper proximity from groups to von Neumann algebras and give various examples of such von Neumann algebras that do not follow from the correct representation of properly proximal groups. The von Neumann algebra generated by f⊗1, f∈L∞(X,µ)andπg,g∈Γ, denoted by L∞(X,µ)oσΓ, is called the construction of the group measure space associated with σ:Γy( X, µ ).
Background on properly proximal groups
An example of emphasis is when the von Neumann algebras are factors, i.e. their center consists only of multiples of the identity, which corresponds to the groups having infinite non-trivial conjugation classes (i.c.c.) and is implied by the fact that the action is quite ergodic and the probability measure is retained. (p.m.p), respectively. Only after Popa's discovery of the deformation/stiffness theory in the early 2000s [Pop07b] have drastic advances been made in the classification of von Neumann algebras in recent years.
Main results
Chapter 2: Properly proximal von Neumann algebras
We also obtain the following dichotomy for subalgebras of von Neumann algebras associated with biexact groups. IfΓ is a biexact group, then every von Neumann subalgebra of LΓ is either properly proximal or has an inclusive direct summand.
Chapter 3: Proper proximality and first ` 2 -Betti numbers
If Λ is correct and contains an unnameable normal subgroup, not properly proximal, then t=1 and ΓyX andΛyY are conjugate. If Γ is exact and not properly proximal, then ΓyX is OE-supersolid, i.e., if Λy(Y,ν) is orbit equivalent to Γy(X,µ), then these two operations are conjugate.
Introduction
The von Neumann algebras associated with properly proximal groups share some of the stiffness properties of those associated with biexact groups. See Section 2.3 for the definition of a boundary piece in the setting of von Neumann algebras.).
Preliminaries
The extended Haagerup tensor product
Strong operator bimodules
Next, we let K1 be the closure of the subspace spanned by π(M)π(M)ξ˜ , and we let K2 be the closure of the subspace spanned by π(N)π(N)η. Suppose that Y ⊂B(H) is an operator space that is both an M-N bimodule and an aM-˜ N bimodule, and suppose that X˜ ⊂Y is both an M-N sub-bimodule and an aM-˜ N sub- bimodule.
Universal representations of normal M-N bimodules
Moreover, if If B⊂A is a C∗-subalgebra such that B is both an M-bimodule and an aM-bimodule and such that the˜ corresponding representations of M andM in M(B)˜ are faithful, then we have a (not unital) recording of von Neumann algebras(BM]M∩BM]˜M˜)∗⊂(AM]M∩AM]˜ M˜)∗.
Relatively compact operators
The space KLX is a closed left ideal in B(L2M), so that M and JMJ are in the space of right multipliers. Since Tφ∗:M→L1Nis is also compact, and sinceφ:N→Mis is bounded, it follows that Tφ∗Tφ is compact as an operator of NtotL1N.
The convolution algebra associated to a von Neumann algebra
Convolution algebras associated to finite von Neumann algebras
It then follows from Wittstock's extension theorem that if W ⊂V is an operatorM,N-subbimodule, then we have an isometric inclusion W[⊂V[. Let M and N be trace von Neumann algebras, assume that we have a uniformly bounded network {xi}i⊂M with kxik2→0 and a uniformly bounded network{µi}i⊂M⊗ConNop.
Mixing operator bimodules
A relative topology on the convolution algebra
We set the X-Y topology to M⊗ehNas the restriction to M⊗ehN of the weak∗ topology in CB(X B(L2N,L2M)Y,B(L2N,L2M)), so that a uniformly bounded net{µi }i⊂M ⊗ehN will converge to µ∈M⊗ehN in the X-Y topology if and only if we have ultraweak convergenceµi(T)→µ(T) for everyT ∈X B(L2N,L2M)Y. From this one then deduces from Lemma 1.2(iv) in [SZ79] that a bounded linear functionalϕ∈(M⊗ehN)∗ is continuous in the X-Y topology if and only ifϕ is continuous in the X-Y topology when is restricted to any bounded set. The same analysis applies to M⊗ConNopasforM⊗ehN, so that in particular a bounded linear functional ϕ∈(M⊗ConNop)∗ is continuous in the X-Y topology if and only ifϕ is continuous in the X-Y topology when it is restricted to any bounded set.
If we now have a uniformly bounded lattice {µi}i ⊂M⊗ConNop, then µi→0 in the X-Y topology (in M⊗ConNop) if and only if for all T ∈XB(L2N,L2M) we have µi( T) → 0 extremely weak and µi∗(JT J)→0 extremely weak.
Relatively mixing bimodules
If{µj}j⊂M⊗ConNopis uniformly bounded such thatµj→0 in the X-Y topology, then by Lemma 2.4.2, for everyϕ∈W∗ we have. Since ϕ|M is normal, we have that µi|M →ϕ|M in the weak topology in M∗ and so by taking convex combinations, we can assume thatkµi|M−ϕ|Mk →0. When we have a net of statesµi∈K(L2M)\Jsuch that{µi}i is uniformly bounded in M⊗ConMop and such thatµi(S)→0for all S∈X, then we haveµi(T)→0.
By Proposition 2.5.1 we have that B(L2N,L2M)X-Y−mix is a strongM-Nbimodule, and it is easy to see that it is also a strongM0-N0bimodule.
Mixing Hilbert bimodules
Let M and N be von Neumann tracheal algebras and assume that X⊂B(L2M) andY⊂ B(L2N) are boundary parts for M and N respectively such that M,JMJ⊂M(X) and N,JNJ⊂M (Y) . Note that equation (2.8) also holds if instead of not In fact, according to the polarization identity, it suffices to check that forξ =η, and if equation (2.8) does not hold, then there exist contractions>0 dhexn∈M such that|hxnξyn,ξi| ≥c.
We have thus established that for every bounded vectorξ ∈H and for every uniformly bounded sequence.
Properly proximal von Neumann algebras
Proper proximality relative to the amenable boundary piece
Let Γ be a group and let π :Γ→U(H) be a universal representation weakly contained in the left regular representation, i.e., π is the restriction of the universal representation of C∗. If Σ is suitable, then the quasi-regular representation `2(Γ/Σ) is weakly contained in the left regular representation and 1Σ is a matrix coefficient. We now fix a formal von Neumann algebra M, and by analogy with above we consider a universal M-MH correspondence weakly contained in the coarse correspondence L2M⊗L2M.
Let H be a universal M-M correspondence that is weakly contained in the coarse correspondence, and letξ,η∈H be bounded vector.
Biexact groups and properly proximal von Neumann algebras
LetΓ be a discrete group, then LΓ is properly proximal relative to Xamenif and only ifΓ is properly proximal relative to Iamen. IfΓ is properly proximal, then clearlyΓ is properly proximal relative to Iamen, but also ifΓ has a normal adaptive subgroupΣCΓso thatΓ/Σ is properly proximal, then it follows from [BIP21] thatΓ is also properly proximal relative to Iamen. Examples of groups and von Neumann algebras that are not properly proximal with respect to the admissible boundary piece are infinite direct products of non-nameable groups or infinite tensor products of non-nameable II1 factors, which follows from the following theorem.
Assume that M is a trace von Neumann algebra and Bn⊂M is a decreasing sequence of von Neumann subalgebras such that each Bn has no admissible summand and such that ∪n(B0n∩M) is ultraweakly dense in M.
Properly proximal actions and crossed products
Then ΓyM is properly proximal with respect to X if and only if NMoΓ(M)yM is properly proximal with respect to X. Suppose that ΓyM is not properly proximal with respect to X and let ϕ be a Γ-equivariant and M-central state on SX(M) with ϕ|M normal and ϕ|. Then ΓyM is properly proximal with respect to Xif if and only if there is no M-centralΓ-invariant stateϕon.
As above, we see that the restriction of ΨtoS(Γ) maps into S(MoΓ), and then the condition ϕ◦Ψ defines aΓ-invariant condition onS(Γ), which shows that Γ is not properly proximal.
Proper proximality and deformation/rigidity theory
Proper proximality via malleable deformations
We also have that the range ofΦ maps into the subspace of invariant operators under the influence of Ad(1⊗ρt). If π:Γ→O(H) is an orthogonal representation, then the Gaussian action associated with π is given by Γ3t 7→απ(t)∈Aut(AH). In particular, if π is a non-name representation, then the restriction of πStoS(H) CΩ∼= H ⊗S(H) is also not, since it is isomorphic to π⊗πS.
If we had aΓ-invariant condition on AH⊗B(S(H) CΩ), restricting it to B(S(H) CΩ) would show that the restriction of the representation πS to S(H) CΩ is available , which would imply that π is an inclusive representation that yields a contradiction.
Proper proximality via closable derivations
A result of Davies and Lindsay in [DL92] shows that D(δ)∩Mis is then again a∗-subalgebra andδ|D(δ)∩M again gives a closable real derivation. The combination of these estimates gives the first part of the lemma, and the second part follows sinceδ is a real derivation. Then K ⊗`2Γ is obviously an LΓ correspondence and we obtain a closable real derivation δ:CΓ→K ⊗`2Γ by setting δ(ut) =c(t)⊗δtfort∈Γ.
Let M be a finite factor, H an M-M correspondence with a real structure that is weakly contained in the coarse correspondence, and suppose δ :L2M→H is a closed real derivation with A= D(δ)∩M.
Introduction
One concrete class of groups satisfying the assumption of Theorem 3.1.1 is the class of one relator groups with at least 3 generators [Gue02, DL07], which were previously not known to be properly proximal. In fact, we first obtain the following convergence superrigidity result for non-properly proximal groups, from which Theorem 3.1.1 follows in combination with [PS12, Corollary 1.2]. Furthermore, since proper proximity and precision are stable under gauge equivalence [IPR19, Oza07], Theorem 3.1.3 also implies the following OE superrigidity result.
Comments on evidence. Let us describe the proof of Theorem 3.1.5, which uses the recently developed notion of proper proximity in [BIP21, IPR19, DKEP22] and Popa's deformation/stiffness theory.
Preliminaries
- Popa’s intertwining-by-bimodules
- Relative amenability
- Mixing subalgebras of finite von Neumann algebras
- Proper proximality
- Boundary pieces
- A bidual characterization
Let be a group that is not properly proximal, then L has no properly proximal direct summand. So for every von Neumann subalgebra N⊂LΓ, either the inclusion N ⊂LΓ is properly proximal to XLΛ, or N has an admissible direct summand. Assume that the inclusion N⊂LΓ is not properly proximal to XLΛ, and let φ :SXLΛ(LΓ)→ hp(LΓ)p,eN pipe ap(LΓ)p-bimodular u.c.p.
Then M is exactly proximal with respect to Xif if and only if there is no central M-state.
From non-proper proximality to relative amenability
Boundary pieces in the bidual
Set ϕ0(T) =ϕ(pT p) +ψ(E˜q(qT q)), which is an M-central state on eS(M) normal to M with support strictly greater than p , which is a contradiction. It is clear that X0 is an M-boundary piece and by inheritance we have eF∈ together with Lemma 3.3.4 thatpt is a projection.
Kpnorι(ProjtΛs∩t0Λs0)pnor, where ProjtΛs∩t0Λs0 denotes the orthogonal projection on sp{tΛs∩t0Λs0}, which is finite-dimensional ift6=t0ors6=s0by the almost malnormality ofΛΓ.
Proof of Proposition 3.3.1
Indeed, each element inX0 can be written as a norm limit of linear extensions consisting of elements of x1Jy1JT Jy2Jx2, where xi,yi∈Cr∗(Γ) andT ∈B(L2B). We have that (ιnor)|M and (ιnor)|JM are faithful normal representations of MandJMJ, respectively, and to eliminate possible confusion, we will denote by ιnor(M)deno(JMJ) the copies of M and JMJ in (B (L2M )]J)∗. Denote me{tk}k≥0⊂Γa representative of the cosetsΓ/Λme0 which is the identity of Γ, i.e., Γ=Fk≥0tkΛ, anduk:=λtk∈U(LΓ).
Therefore, we conclude that φ0 is a continuous norm∗-homomorphism from A0toq⊥KS˜(M) and thus extends to the C∗-algebraA:=A0k·k.
From relative amenability to rigidity
Let αt =σUt andβ =σV be the associated automorphisms of AH⊕H ∼=AH⊗AH, and both extend to Aut((AH⊗AH)oσπ⊗σπΓ), still denoted by αt andβ, asV andUt commute with (π( ⊕ϕ) Γ) andσπ⊕π =σπ⊗σπ. Suppose there exists another trace von Neumann algebra(M,˜ τ)˜ such that M⊂M and˜ τ˜|M=τ, and a net of trace-preserving automorphisms{θt}t∈R⊂Aut(M )˜ such thatθt |B∈Aut(B), and such thatθt|M→idM in the point-k · k2topology, if t→0. Suppose N⊂pM p is a von Neumann subalgebra, for some p∈P(M), with no adaptable direct call, such that N is admissible.
Since AH is abelian, N is diffuse, and Qis of type II1, the statementQ≺M LΓ follows directly from [Bou12, Theorem 3.4] and Proposition 3.4.2.
Proofs of main theorems
BIP21] R´emi Boutonnet, Adrian Ioana og Jesse Peterson, Properly proximal groups and their von Neumann algebras, Ann. DKEP22] Changying Ding, Srivatsav Kunnawalkam Elayavalli og Jesse Peterson, Properly proximal von Neumann algebras, 2022, arXiv:2204.00517. MP03] Nicolas Monod og Sorin Popa, On co-amenability for groups og von Neumann algebras, C.
Pop07b] ,Deformation and stiffness for group actions and von Neumann algebras, International Congress of Mathematicians.
