scholarly journals An inner amenable group whose von Neumann algebra does not have property Gamma

2012 ◽  
Vol 208 (2) ◽  
pp. 389-394 ◽  
Author(s):  
Stefaan Vaes
Keyword(s):  
Von Neumann Algebra ◽  
Amenable Group ◽  
Von Neumann

On groups of automorphisms of operator algebras

1977 ◽  
Vol 81 (2) ◽  
pp. 237-243 ◽  
Author(s):  
J. Moffat
Keyword(s):  
Von Neumann Algebra ◽  
Separable Hilbert Space ◽  
Operator Algebras ◽  
Amenable Group ◽  
Locally Compact Abelian Group ◽  
Von Neumann ◽  
Weakly Continuous ◽  
Continuous Unitary Representation ◽  
Inner Automorphisms ◽  
Necessary And Sufficient

In section 3 we shall prove the following results: Let G be a separable locally compact abelian group, R a von Neumann algebra acting on a separable Hilbert space, and α a weakly continuous representation of G by inner *-automorphisms of R, say α(g) = ad Wg with Wg ∈ U(R). Then there is a weakly continuous unitary representation of G, by unitaries in R, implementing α if and only if the Wg's commute with each other. The result was motivated by the proof of (7), theorem 1. Suppose now Gis a discrete amenable group of *-automorphisms of a countably decomposable von Neumann algebra R. In section 3 we give a necessary and sufficient condition for the existence of a faithful normal G-invariant state on R. This generalizes a result of Hajian and Kakutani on invariant measures (2).

scholarly journals Inner amenability, property Gamma, McDuff factors and stable equivalence relations

2017 ◽  
Vol 38 (7) ◽  
pp. 2618-2624 ◽  
Author(s):  
TOBE DEPREZ ◽  
STEFAAN VAES
Keyword(s):  
Probability Measure ◽  
Von Neumann Algebra ◽  
Amenable Group ◽  
Countable Group ◽  
Acta Math ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Von Neumann ◽  
Stable Equivalence ◽  
Inner Amenability

We say that a countable group $G$ is McDuff if it admits a free ergodic probability measure preserving action such that the crossed product is a McDuff $\text{II}_{1}$ factor. Similarly, $G$ is said to be stable if it admits such an action with the orbit equivalence relation being stable. The McDuff property, stability, inner amenability and property Gamma are subtly related and several implications and non-implications were obtained in Effros [Property $\unicode[STIX]{x1D6E4}$ and inner amenability. Proc. Amer. Math. Soc.47 (1975), 483–486], Jones and Schmidt [Asymptotically invariant sequences and approximate finiteness. Amer. J. Math.109 (1987), 91–114], Vaes [An inner amenable group whose von Neumann algebra does not have property Gamma. Acta Math.208 (2012), 389–394], Kida [Inner amenable groups having no stable action. Geom. Dedicata173 (2014), 185–192] and Kida [Stability in orbit equivalence for Baumslag–Solitar groups and Vaes groups. Groups Geom. Dyn.9 (2015), 203–235]. We complete the picture with the remaining implications and counterexamples.

scholarly journals A computational approach to the Thompson group F

2015 ◽  
Vol 25 (03) ◽  
pp. 381-432 ◽  
Author(s):  
Søren Haagerup ◽  
Uffe Haagerup ◽  
Maria Ramirez-Solano
Keyword(s):  
Von Neumann Algebra ◽  
Regular Representation ◽  
Amenable Group ◽  
Upper Bounds ◽  
Computational Approach ◽  
Computational Results ◽  
Von Neumann ◽  
Spectral Distributions ◽  
Group Von Neumann Algebra ◽  
Thompson Group

Let F denote the Thompson group with standard generators A = x0, B = x1. It is a long standing open problem whether F is an amenable group. By a result of Kesten from 1959, amenability of F is equivalent to [Formula: see text] and to [Formula: see text] where in both cases the norm of an element in the group ring ℂF is computed in B(ℓ2(F)) via the regular representation of F. By extensive numerical computations, we obtain precise lower bounds for the norms in (i) and (ii), as well as good estimates of the spectral distributions of (I+A+B)*(I+A+B) and of A+A-1+B+B-1 with respect to the tracial state τ on the group von Neumann Algebra L(F). Our computational results suggest, that [Formula: see text] It is however hard to obtain precise upper bounds for the norms, and our methods cannot be used to prove non-amenability of F.

scholarly journals Outer actions of a discrete amenable group on approximately finite dimensional factors II, the III$_{\lambda}$-case, $\lambda\neq 0$

2007 ◽  
Vol 100 (1) ◽  
pp. 75 ◽  
Author(s):  
Yoshikazu Katayama ◽  
Masamichi Takesaki
Keyword(s):  
Abelian Group ◽  
Cohomology Group ◽  
Von Neumann Algebra ◽  
Amenable Group ◽  
Classification Result ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Outer Action ◽  
The One ◽  
Abelian Von Neumann Algebra

To study outer actions $\alpha$ of a group $G$ on a factor $\mathcal M$ of type $\mathrm{III}_\lambda$, $0<\lambda<1$, we study first the cohomology group of a group with the unitary group of an abelian von Neumann algebra as a coefficient group and establish a technique to reduce the coefficient group to the torus $\mathsf T$ by the Shapiro mechanism based on the groupoid approach. We then show a functorial construction of outer actions of a countable discrete amenable group on an AFD factor of type $\mathrm{III}_\lambda$, sharpening the result in [17, §4]. The periodicity of the flow of weights on a factor $\mathcal M$ of type $\mathrm{III}_\lambda$ allows us to introduce an equivariant commutative square directly related to the discrete core. But this makes it necessary to introduce an enlarged group $\mathrm{Aut}(\mathcal M)_{m}$ relative to the modulus homomorphism $m=\mod\colon \mathrm{Aut}(\mathcal M)\to \mathsf R/T'\mathsf Z$. We then discuss the reduced modified HJR-exact sequence, which allows us to describe the invariant of outer action $\alpha$ in a simpler form than the one for a general AFD factor: for example, the cohomology group $H_{m,s}^{out}(G,N,\mathsf T)$ of modular obstructions is a compact abelian group. Making use of these reductions, we prove the classification result of outer actions of $G$ on an AFD factor $\mathcal M$ of type $\mathrm{III}_{\lambda}$.

A mean ergodic theorem of an amenable group action

2014 ◽  
Vol 17 (01) ◽  
pp. 1450003 ◽  
Author(s):  
Anilesh Mohari
Keyword(s):  
Ergodic Theorem ◽  
Group Action ◽  
Von Neumann Algebra ◽  
Amenable Group ◽  
Ergodic Theorems ◽  
Von Neumann ◽  
Mean Ergodic Theorem ◽  
Faithful Normal State ◽  
Contractive Maps ◽  
Dual Banach Space

We consider a sequence of weak Kadison–Schwarz maps τn on a von-Neumann algebra ℳ with a faithful normal state ϕ sub-invariant for each (τn, n ≥ 1) and use a duality argument to prove strong convergence of their pre-dual maps when their induced contractive maps (Tn, n ≥ 1) on the GNS space of (ℳ, ϕ) are strongly convergent. The result is applied to deduce improvements of some known ergodic theorems and Birkhoff's mean ergodic theorem for any locally compact second countable amenable group action on the pre-dual Banach space ℳ*.

scholarly journals INNER AMENABLE GROUPOIDS AND CENTRAL SEQUENCES

2020 ◽  
Vol 8 ◽  
Author(s):  
YOSHIKATA KIDA ◽  
ROBIN TUCKER-DROB
Keyword(s):  
Von Neumann Algebra ◽  
Analogous Result ◽  
Amenable Group ◽  
Equivalence Relations ◽  
Full Group ◽  
Discrete Probability ◽  
Von Neumann ◽  
Central Sequence ◽  
Central Sequences ◽  
Inner Amenability

We introduce inner amenability for discrete probability-measure-preserving (p.m.p.) groupoids and investigate its basic properties, examples, and the connection with central sequences in the full group of the groupoid or central sequences in the von Neumann algebra associated with the groupoid. Among other things, we show that every free ergodic p.m.p. compact action of an inner amenable group gives rise to an inner amenable orbit equivalence relation. We also obtain an analogous result for compact extensions of equivalence relations that either are stable or have a nontrivial central sequence in their full group.

scholarly journals Local rigidity for actions of Kazhdan groups on noncommutative Lp-spaces

2015 ◽  
Vol 26 (08) ◽  
pp. 1550064
Author(s):  
Bachir Bekka
Keyword(s):  
Von Neumann Algebra ◽  
Regular Representation ◽  
Von Neumann ◽  
Lp Spaces ◽  
Local Rigidity ◽  
Left Regular Representation ◽  
Left Regular ◽  
Property T ◽  
Finite Factor ◽  
Finite Index Subgroup

Let Γ be a discrete group and 𝒩 a finite factor, and assume that both have Kazhdan's Property (T). For p ∈ [1, +∞), p ≠ 2, let π : Γ →O(Lp(𝒩)) be a homomorphism to the group O(Lp(𝒩)) of linear bijective isometries of the Lp-space of 𝒩. There are two actions πl and πr of a finite index subgroup Γ+ of Γ by automorphisms of 𝒩 associated to π and given by πl(g)x = (π(g) 1)*π(g)(x) and πr(g)x = π(g)(x)(π(g) 1)* for g ∈ Γ+ and x ∈ 𝒩. Assume that πl and πr are ergodic. We prove that π is locally rigid, that is, the orbit of π under O(Lp(𝒩)) is open in Hom (Γ, O(Lp(𝒩))). As a corollary, we obtain that, if moreover Γ is an ICC group, then the embedding g ↦ Ad (λ(g)) is locally rigid in O(Lp(𝒩(Γ))), where 𝒩(Γ) is the von Neumann algebra generated by the left regular representation λ of Γ.

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