scholarly journals Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule

2004 ◽  
Vol 95 (1) ◽  
pp. 124 ◽  
Author(s):  
Volker Runde
Keyword(s):  
Abelian Subgroup ◽  
Von Neumann Algebras ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Von Neumann ◽  
Connes Amenability ◽  
Definition Of ◽  
Dual Banach Algebra

Let $\mathcal A$ be a dual Banach algebra with predual $\mathcal A_*$ and consider the following assertions: (A) $\mathcal A$ is Connes-amenable; (B) $\mathcal A$ has a normal, virtual diagonal; (C) $\mathcal A_*$ is an injective $\mathcal A$-bimodule. For general $\mathcal A$, all that is known is that (B) implies (A) whereas, for von Neumann algebras, (A), (B), and (C) are equivalent. We show that (C) always implies (B) whereas the converse is false for $\mathcal A = M(G)$ where $G$ is an infinite, locally compact group. Furthermore, we present partial solutions towards a characterization of (A) and (B) for $\mathcal A = B(G)$ in terms of $G$: For amenable, discrete $G$ as well as for certain compact $G$, they are equivalent to $G$ having an abelian subgroup of finite index. The question of whether or not (A) and (B) are always equivalent remains open. However, we introduce a modified definition of a normal, virtual diagonal and, using this modified definition, characterize the Connes-amenable, dual Banach algebras through the existence of an appropriate notion of virtual diagonal.

Decomposability of von Neumann Algebras and the Mazur Property of Higher Level

2006 ◽  
Vol 58 (4) ◽  
pp. 768-795 ◽  
Author(s):  
Zhiguo Hu ◽  
Matthias Neufang
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Group Structure ◽  
Cardinal Number ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Von Neumann ◽  
Topological Centre

AbstractThe decomposability number of a von Neumann algebra ℳ (denoted by dec(ℳ)) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in ℳ. In this paper, we explore the close connection between dec(ℳ) and the cardinal level of the Mazur property for the predual ℳ* of ℳ, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group G as the group algebra L1(G), the Fourier algebra A(G), the measure algebra M(G), the algebra LUC(G)*, etc. We show that for any of these von Neumann algebras, say ℳ, the cardinal number dec(ℳ) and a certain cardinal level of the Mazur property of ℳ* are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of G: the compact covering number κ(G) of G and the least cardinality ᙭(G) of an open basis at the identity of G. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra A(G)**.

scholarly journals Amenability of groupoids and asymptotic invariance of convolution powers

2021 ◽  
pp. 69-92
Author(s):  
Theo Bühler ◽  
Vadim Kaimanovich
Keyword(s):  
Random Walk ◽  
Locally Compact Group ◽  
Probability Measures ◽  
Locally Compact ◽  
Von Neumann ◽  
Original Definition ◽  
Convolution Powers ◽  
Measure Class ◽  
Definition Of

The original definition of amenability given by von Neumann in the highly non-constructive terms of means was later recast by Day using approximately invariant probability measures. Moreover, as it was conjectured by Furstenberg and proved by Kaimanovich–Vershik and Rosenblatt, the amenability of a locally compact group is actually equivalent to the existence of a single probability measure on the group with the property that the sequence of its convolution powers is asymptotically invariant. In the present article we extend this characterization of amenability to measured groupoids. It implies, in particular, that the amenability of a measure class preserving group action is equivalent to the existence of a random environment on the group parameterized by the action space, and such that the tail of the random walk in almost every environment is trivial.

Ergodic actions of group extensions on von Neumann algebras

1989 ◽  
Vol 112 (1-2) ◽  
pp. 71-112
Author(s):  
Klaus Thomsen
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Fixed Point Algebra ◽  
Atomic Centre

SynopsisWe consider automorphic actions on von Neumann algebras of a locally compact group E given as a topological extension 0 → A → E → G → 0, where A is compact abelian and second countable. Motivated by the wish to describe and classify ergodic actions of E when G is finite, we classify (up to conjugacy) first the ergodic actions of locally compact groups on finite-dimensional factors and then compact abelian actions with the property that the fixed-point algebra is of type I with atomic centre. We then handle the case of ergodic actions of E with the property that the action is already ergodic when restricted to A, and then, as a generalisation, the case of (not necessarily ergodic) actions of E with the property that the restriction to A is an action with abelian atomic fixed-point algebra. Both these cases are handled for general locally compact-countable G. Finally, we combine the obtained results to classify the ergodic actions of E when G is finite, provided that either the extension is central and Hom (G, T) = 0, or G is abelian and either cyclic or of an order not divisible by a square.

scholarly journals Johnson pseudo-Connes amenability of dual Banach algebras

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 551-559
Author(s):  
Amir Sahami ◽  
Seyedeh Shariati ◽  
Abdolrasoul Pourabbas
Keyword(s):  
Compact Group ◽  
Finite Index ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Hereditary Properties ◽  
Connes Amenability

We introduce the notion of Johnson pseudo-Connes amenability for dual Banach algebras. We study the relation between this new notion with the various notions of Connes amenability like Connes amenability, approximate Connes amenability and pseudo Connes amenability. We also investigate some hereditary properties of this new notion. We prove that for a locally compact group G,M(G) is Johnson pseudo-Connes amenable if and only if G is amenable. Also we show that for every non-empty set I,MI(C) under this new notion is forced to have a finite index. Finally, we provide some examples of certain dual Banach algebras and we study their Johnson pseudo-Connes amenability.

scholarly journals Approximate Connes-biprojectivity of dual Banach algebras

2020 ◽  
pp. 2150025
Author(s):  
A. Sahami ◽  
S. F. Shariati ◽  
A. Pourabbas
Keyword(s):  
Compact Group ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Convolution Product ◽  
Locally Compact ◽  
Triangular Banach Algebras ◽  
Connes Amenability

In this paper, we introduce a notion of approximate Connes-biprojectivity for dual Banach algebras. We study the relation between approximate Connes-biprojectivity, approximate Connes amenability and [Formula: see text]-Connes amenability. We propose a criterion to show that certain dual triangular Banach algebras are not approximately Connes-biprojective. Next, we show that for a locally compact group [Formula: see text], the Banach algebra [Formula: see text] is approximately Connes-biprojective if and only if [Formula: see text] is amenable. Finally, for an infinite commutative compact group [Formula: see text], we show that the Banach algebra [Formula: see text] with convolution product is approximately Connes-biprojective, but it is not Connes-biprojective.

scholarly journals On Group Von Neumann Algebras with Vector-Valued Functions

2018 ◽  
Vol 14 (1) ◽  
pp. 7596-7614
Author(s):  
Julien Esse Atto ◽  
Victor Kofi Assiamoua
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Regular Representation ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Von Neumann ◽  
Left Regular Representation ◽  
Left Regular ◽  
Vector Valued Functions ◽  
Vector Valued

Let G be a locally compact group equipped with a normalized Haar measure , A(G) the Fourier algebraof G and V N(G) the von Neumann algebra generated by the left regular representation of G. In this paper, we introduce the space V N(G;A) associated with the Fourier algebra A(G;A) for vector-valued functions on G, where A is a H-algebra. Some basic properties are discussed in the category of Banach space, and alsoin the category of operator space.

Invariant Means on a Class of von Neumann Algebras Related to Ultraspherical Hypergroups II

2017 ◽  
Vol 60 (2) ◽  
pp. 402-410
Author(s):  
N. Shravan Kumar
Keyword(s):  
Compact Group ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Von Neumann ◽  
Invariant Means

AbstractLet K be an ultraspherical hypergroup associated with a locally compact group G and a spherical projector π and let VN(K) denote the dual of the Fourier algebra A(K) corresponding to K. In this note, we show that the set of invariant means on VN(K) is singleton if and only if K is discrete. Here K need not be second countable. We also study invariant means on the dual of the Fourier algebra A0(K), the closure of A(K) in the cb-multiplier norm. Finally, we consider generalized translations and generalized invariant means.

scholarly journals A Duality of Locally Compact Groups That Does Not Involve the Haar Measure

2015 ◽  
Vol 116 (2) ◽  
pp. 250 ◽  
Author(s):  
Yulia Kuznetsova
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
C Algebras ◽  
Commutative Algebras

We present a simple and intuitive framework for duality of locally compacts groups, which is not based on the Haar measure. This is a map, functorial on a non-degenerate subcategory, on the category of coinvolutive Hopf $C^*$-algebras, and a similar map on the category of coinvolutive Hopf-von Neumann algebras. In the $C^*$-version, this functor sends $C_0(G)$ to $C^*(G)$ and vice versa, for every locally compact group $G$. As opposed to preceding approaches, there is an explicit description of commutative and co-commutative algebras in the range of this map (without assumption of being isomorphic to their bidual): these algebras have the form $C_0(G)$ or $C^*(G)$ respectively, where $G$ is a locally compact group. The von Neumann version of the functor puts into duality, in the group case, the enveloping von Neumann algebras of the algebras above: $C_0(G)^{**}$ and $C^*(G)^{**}$.

Voiculescu amenable Hopf von Neumann algebras with applications

2016 ◽  
Vol 15 (06) ◽  
pp. 1650079 ◽  
Author(s):  
Fatemeh Akhtari ◽  
Rasoul Nasr-Isfahani
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Continuous Functions ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann

For a Hopf von Neumann algebra [Formula: see text], we give a fixed point characterization of Voiculescu amenability of [Formula: see text] in terms of modules over [Formula: see text]. As a consequence, we present some descriptions for amenability of locally compact groups in terms of certain associated Hopf von Neumann algebras. We finally apply this result to some modules of continuous functions on a multiplicative subsemigroup of [Formula: see text].

Sign in / Sign up

Export Citation Format

Share Document