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 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.

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 On Ultraproduct Embeddings and Amenability for Tracial von Neumann Algebras

2020 ◽  
Author(s):  
Scott Atkinson ◽  
Srivatsav Kunnawalkam Elayavalli
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Analogous Result ◽  
Equivalence Classes ◽  
Embedding Problem ◽  
Unitary Equivalence ◽  
Von Neumann ◽  
Sofic Groups

Abstract We define the notion of self-tracial stability for tracial von Neumann algebras and show that a tracial von Neumann algebra satisfying the Connes embedding problem (CEP) is self-tracially stable if and only if it is amenable. We then generalize a result of Jung by showing that a separable tracial von Neumann algebra that satisfies the CEP is amenable if and only if any two embeddings into $R^{\mathcal{U}}$ are ucp-conjugate. Moreover, we show that for a II$_1$ factor $N$ satisfying CEP, the space $\mathbb{H}$om$(N, \prod _{k\to \mathcal{U}}M_k)$ of unitary equivalence classes of embeddings is separable if and only $N$ is hyperfinite. This resolves a question of Popa for Connes embeddable factors. These results hold when we further ask that the pairs of embeddings commute, admitting a nontrivial action of $\textrm{Out}(N\otimes N)$ on ${\mathbb{H}}\textrm{om}(N\otimes N, \prod _{k\to \mathcal{U}}M_k)$ whenever $N$ is non-amenable. We also obtain an analogous result for commuting sofic representations of countable sofic groups.

scholarly journals Operator algebras related to Thompson's group F

2005 ◽  
Vol 79 (2) ◽  
pp. 231-241 ◽  
Author(s):  
Paul Jolissaint
Keyword(s):  
Cyclic Group ◽  
Commutator Subgroup ◽  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Natural Action ◽  
Von Neumann ◽  
Ratio Set ◽  
Thompson’S Group ◽  
Central Sequences ◽  
Abelian Von Neumann Algebra

AbstractLet F′ be the commutator subgroup of F and let Γ0 be the cyclic group generated by the first generator of F. We continue the study of the central sequences of the factor L(F′), and we prove that the abelian von Neumann algebra L(Γ0) is a strongly singular MASA in L(F). We also prove that the natural action of F on [0, 1] is ergodic and that its ratio set is {0} ∪ {2k; k ∞ Z}.

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 $\mathrm{III}_{\lambda}$.

scholarly journals Inner amenability for groups and central sequences in factors

2014 ◽  
Vol 36 (4) ◽  
pp. 1106-1129 ◽  
Author(s):  
IONUT CHIFAN ◽  
THOMAS SINCLAIR ◽  
BOGDAN UDREA
Keyword(s):  
Measure Space ◽  
Negative Curvature ◽  
Mapping Class ◽  
Discrete Groups ◽  
Class Groups ◽  
Curvature Condition ◽  
Von Neumann ◽  
Group Measure ◽  
Central Sequences ◽  
Inner Amenability

We show that a large class of i.c.c., countable, discrete groups satisfying a weak negative curvature condition are not inner amenable. By recent work of Hull and Osin [Groups with hyperbolically embedded subgroups. Algebr. Geom. Topol.13 (2013), 2635–2665], our result recovers that mapping class groups and $\text{Out}(\mathbb{F}_{n})$ are not inner amenable. We also show that the group-measure space constructions associated to free, strongly ergodic p.m.p. actions of such groups do not have property Gamma of Murray and von Neumann [On rings of operators IV. Ann. of Math. (2) 44 (1943), 716–808].

scholarly journals A VON NEUMANN ALGEBRA CHARACTERIZATION OF PROPERTY (T) FOR GROUPOIDS

2018 ◽  
Vol 108 (3) ◽  
pp. 363-386
Author(s):  
MARTINO LUPINI
Keyword(s):  
Probability Measure ◽  
Von Neumann Algebra ◽  
Discrete Probability ◽  
Von Neumann ◽  
Relative Property ◽  
Property T ◽  
Algebra Characterization

For an arbitrary discrete probability-measure-preserving groupoid $G$, we provide a characterization of property (T) for $G$ in terms of the groupoid von Neumann algebra $L(G)$. More generally, we obtain a characterization of relative property (T) for a subgroupoid $H\subset G$ in terms of the inclusions $L(H)\subset L(G)$.

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