The first successes in this direction were achieved by Popa and include a number of surprising results: examples of factors with a trivial basis set [Po01] answering an old Kadison open problem [K67] (see [Ge03, Problem 3] ) ;. In fact, no clear calculation of the basis set of any property factor (T) exists in the current literature. In Section 5 we make progress on this problem by providing examples of properties (T) of iccGgroups whose L(G) factors have trivial basis sets.
We also show that products of finite groups in class V give rise to property (T) type II1factors with trivial fundamental group.
Definitions
In this chapter, we introduce and gather facts about several topics in von Neumann algebras that will be used throughout this thesis. The Avon Neumann algebra (over a Hilbert spaceH ) is a ∗-subalgebra of B(H ) which contains 1 and is closed in the topology of the weak operator. Since the subalgebras are convex, it follows from Corollary 2.1.4 that the von Neumann algebras are also closed under the strong operator topology.
But since A is self-adjoint it also follows that for alla∈Awe we have pa−(a∗p)∗= (pa∗p)∗=pap=ap, and thus∈A0.
Examples
- Abelian von Naumann algebra
- Tracial von Neumann algebra
- Group von Neumann algebra
At least LΓ is a von Neumann algebra, which also generates λ(Γ), RΓ is a von Neumann algebra, which also generates ρ(Γ). Since RΓ is generated by z ρ(Γ), it is sufficient to show that λ(ξ) commutes with ρ(g) for all g∈Γ, which is clear from the effect that λ(ξ) is the left convolution of z ξ in ρ( g) right convolution z δg. If Γ is a harmful discrete group, the von Neumann algebra LΓ, defined above, is called the group von Neumann algebra associated with the group Γ.
Notations and Terminology
IfM is a tracial von Neumann algebra andGyσM is a trace-preserving action, we denote by M oσG the corresponding cross product of the Neumann algebra [MvN37]. For each subset K⊆G we denote with PMK the orthogonal projection of the Hilbert space L2(MoG) onto the closed linear span of {xug|x∈M,g∈K}. All groups covered in this article are countable and are indicated by capital letters A, B, G, H, Q, N, M, etc.
Given groups Q, N and an action Qyσ N by automorphisms, we denote by NoσQ the corresponding semidirect product group.
Popa’s Intertwining Techniques
Po03] Let(M,τ) be a separable tracial von Neumann algebra and letP,Q⊆M be (not necessarily unitary) von Neumann subalgebras. Suppose H 6 G is an almost abnormal subgroup and let GyN be a trace-preserving action on a finite von Neumann algebra N. Assume that H1,H26G are groups, let GyN be a trace-preserving action on a tracial von Neumann algebraN and give with M =N oG denotes the corresponding crossed product.
Let M be a finite von Neumann algebra and let N be a type II1 factor, with N ⊆M a unit inclusion.
Height of elements in group von Neumann algebras
Since N is a type II1 factor, we can find orthogonal projections pi∈P(N), and unitariesui∈U(N)(fori=2,· · ·,n) like this. Next, we highlight a new situation where it is possible to check the lower bound of the height of unit elements in the context of crossed product von Neumann algebras arising from group actions of automorphisms without non-trivial stabilizers. Our result and its proof are reminiscent of the earlier powerful techniques for Bernoulli actions introduced in [IPV10, Theorem 5.1] (see also [Io11, Theorem 6.1]) and their recent counterparts for the Rips constructions [CDK19, Theorem 5.1].
Let G and H be countable groups and let σ :G→Aut(H) be the action of automorphisms for which there exists a scalar c>0 corresponding to|StabG(h)|
In the direction of this first one, we note that since A ≺sM N then by [Va10a, Lemma 2.5] for every ε there exists a finite set S⊆K, so that for all c∈U(A) holds. Then we also claim that for every finite set S⊂G and every ε >0 there exists b∈U(A) such that. If we now take a=PNS(b)−EL(G)(PNS(b)) and apply (2.23), we get that the last inequality above is less than.
Since this is true for all g∈G, we get the desired conclusion if we let ε >0 be small enough.
A class of groups based on Belegradek-Osin Rips construction
Concrete examples of semidirect product groups in class Scan can be obtained if the starting groups Qi are any uniform lattices iSp(n,1)whenn≥2. Finally, we conclude this section with a folkloric lemma related to the computation of centralizers of elements in products of hyperbolic groups. For any e6=g∈Q, the centralizer CQ(g) is of one of the following forms: A, A×Q2 or Q1×A, where A is an inclusive group.
To obtain our conclusion, it is therefore sufficient to show that for every gi∈Qie either CQi(gi) =Qi or CQi(gi) is an elementary group. However, this is immediate when we note that for everygi6=e the centralizer satisfies CQi(gi)6EQi(gi), where EQi(gi) is maximal elementary subgroup containing gi of the torsion-free icc hyperbolic groupQi, see for example [Ol91].
A class of groups V arising from Valette’s examples
An impressive milestone in the classification of von Neumann algebras was the emergence in the last decade of the first examples of groupsG that can be completely reconstructed from their von Neumann algebrasL(G), i.e. W∗-superrigid groups[ IPV10, BV12, CI17]. This naturally leads to a broad and independent study, specifically identifying canonical group algebraic features of a group that are passed on to von Neumann algebra. Furthermore, in the current literature there is an almost complete lack of examples of algebraic features that occur in a feature group (T) and that are recognizable at the von Neumann algebraic level.
Therefore, in order to successfully construct (T)W∗-supersolid property groups via a strategy similar to [IPV10, CI17], we believe it is imperative to first identify a comprehensive list of algebraic property groups of properties (T) that survive von Neumann's algebraic structure. Then for a sufficiently large family of groups Q we show that the semidirect product property of G is an algebraic property fully recoverable from the von Neumann algebraic regime. Furthermore, one can find a multiplicative character η :Q→ T, a group isomorphism δ :Q→H, a unitary w∈L(Λ) and ∗-isomorphisms Θi:L(Ni)→L(Ki) such that that for all xi∈L(Ni) and g∈Q we have.
From another perspective, our theorem can also be seen as a von Neumann algebraic superstiffness result concerning conjugation of actions on non-commutative von Neumann algebras. Following [CIK13, notation 3.3], we now consider the von Neumann algebraic embedding corresponding to πi, i.e. Πi:L(Gi)→L(Gi)⊗¯L(Fi) given by Πi(ug) =ug⊗vπi(g)for alleg ∈G;. FixA ⊂Π(N1) any diffusely admissible von Neumann subalgebra. On the other hand, if case a) above were to hold for all A's.
Therefore we have in particular that vv∗∈B0∩qMq⊆L(Gˆk×Hk) and thus relation (4.3) implies that Bvv∗=vNiv∗⊆L(Gˆk×Hk). Finally, we proved that vL(Φ)zv∗⊆rL(Q)rand for all6zand f 6dprojections in the factorL(Φ)zso thatτ(f)>τ(e) we haveL(Φ)ev∗=roL(Q)rowherero6r = vzv∗. Note that since Q<Γ= (N1×N2)oQ is almost nonnormal, then we have the following property: for every sequence L(Q)3xn→0 weak and every x,y∈M such thatEL(Q)(x ) = EL( Q)(y) =0 we have.
Then we show that in (4.33) we can choose z∈Z(L(Φ)) at most with the property that we have for every projection ∈Z(L(Φ)z⊥).
Proof of Theorem 4.0.1
Reconstruction of the Acting Group Q
Astnnormalizes∆(A), then you can see that. 4.41) Since the stabilizer sizes are uniformly limited, we get a contradiction if ε >0 is arbitrarily small.
Reconstruction of a Core Subgroup and its Product Feature
For example, one can easily see that the von Neumann algebras covered by this theorem are non-isomorphic to those arising from any irreducible lattice in higher-ranking Lie groups. Indeed, if such a lattice satisfies L(Γ)∼=L(Λ), then Theorem 4.0.1 would imply that Λ must contain an infinite normal subgroup of infinite index, which contradicts Margulis' normal subgroup theorem. 2) Although it is well known that there are countless non-isomorphic group II1 factors with property (T) [Po07], little is known about producing concrete examples of such families. For example, in the statement one can simply vary Qi in any infinite family of non-isomorphic uniform lattices in Sp(n,1) for anyn6=2.. the other families our consist of factors which are not solid, do not admit tensor- decompositions [CdSS17], and do not have Cartan subalgebras, [CIK13]. 3) We note that theorem 4.0.1 still holds if instead of Γ= (N1×N2)o(Q1×Q2) one considers any finite index subgroup ofΓof the formΓs,r = (N1×N2)o(Qs1×Qr2 ) )6Γ, where Qs16Q1 and Qr26Q2 are arbitrary finite index subgroups.
It can be verified that these groups still possess all the algebraic/geometric properties used in the proof of Theorem 4.0.1 (including the fact that N1oQs1 is hyperbolic with respect to Qs1 and N1oQr2 is hyperbolic with respect to Qr2) and therefore all von Neumann's algebraic arguments in the proof. apply Theorem 4.0.1 verbatim. In this section we continue this investigation by showing in particular that these factors also have a trivial fundamental group (see Theorem 5.1.6 and Corollary 5.1.10). Note that A ⊆Θ(N) and B ⊆ pMp are amplifications of real crossed product inclusions. Also according to part d) in Theorem 3.1.1A it is a regular irreducible subfactor of Θ(N) =pMp, while Bi is a quasi-regular irreducible subfactor of pMp(asQN pMp(pBp)00= pQN M(L(M) )p).
Furthermore, from Theorem 3.1.1 we can see that the action σ:P→Aut(M) satisfies all the conditions in the hypothesis of Theorem 2.1.31 and thus using the conclusion of the same theorem we obtain. Finally, continuing as in the proof of [CDK19, Theorem 5.1] it can be further shown that the isomorphism Ψ arises from a tensor of ∗-isomorphismsΦi:L(Ni)→L(Mi). We refer the reader to Section 3.2 for elementary properties of these groups and their von Neumann algebras.
According to stiffness results for von Neumann algebras arising from mixing extensions of profinite group actions in probability spaces, https://arxiv.org/abs/1903.07143. Khan, Some applications of group-theoretic tear constructions to the classification of von Neumann algebras, https://arxiv.org/pdf/1911.11729. Jones, Property (T) for von Neumann algebras, Bull. French) [Factor Classification] Operator Algebra and Applications, Part 2 (Kingston, Ont., 1980), pp.
Popa, Deformation and rigidity for group actions and von Neumann algebras, International Congress of Mathematicians.
