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.

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.

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.

A NOTE ON THE NORMS OF TRANSITION OPERATORS ON LAMPLIGHTER GRAPHS AND GROUPS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1261-1272 ◽  
Author(s):  
WOLFGANG WOESS
Keyword(s):  
Random Walk ◽  
Wreath Products ◽  
Locally Compact Group ◽  
Operator Matrix ◽  
Locally Compact ◽  
Finitely Generated Groups ◽  
Group Of Isometries ◽  
Transition Operators ◽  
Special Case ◽  
Product Of Groups

Let L≀X be a lamplighter graph, i.e., the graph-analogue of a wreath product of groups, and let P be the transition operator (matrix) of a random walk on that structure. We explain how methods developed by Saloff-Coste and the author can be applied for determining the ℓp-norms and spectral radii of P, if one has an amenable (not necessarily discrete or unimodular) locally compact group of isometries that acts transitively on L. This applies, in particular, to wreath products K≀G of finitely-generated groups, where K is amenable. As a special case, this comprises a result of Żuk regarding the ℓ2-spectral radius of symmetric random walks on such groups.

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 Continuity of Convolution of Test Functions on Lie Groups

2014 ◽  
Vol 66 (1) ◽  
pp. 102-140
Author(s):  
Lidia Birth ◽  
Helge Glöckner
Keyword(s):  
Continuous Functions ◽  
Locally Convex Space ◽  
Locally Compact Group ◽  
Locally Convex Spaces ◽  
Test Functions ◽  
Bilinear Map ◽  
Locally Compact ◽  
Compactly Supported ◽  
Locally Convex

AbstractFor a Lie group G, we show that the map taking a pair of test functions to their convolution, is continuous if and only if G is σ-compact. More generally, consider with t ≤ r + s, locally convex spaces E1, E2 and a continuous bilinear map b : E1 × E2 → F to a complete locally convex space F. Let be the associated convolution map. The main result is a characterization of those (G; r; s; t; b) for which β is continuous. Convolution of compactly supported continuous functions on a locally compact group is also discussed as well as convolution of compactly supported L1-functions and convolution of compactly supported Radon measures.

Continuous Ergodic Extensions and Fibre Bundles

1978 ◽  
Vol 30 (02) ◽  
pp. 373-391 ◽  
Author(s):  
Robert J. Zimmer
Keyword(s):  
Real Line ◽  
Lebesgue Space ◽  
Transformation Group ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Von Neumann ◽  
Fibre Bundles ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Measure Preserving Transformation

If a locally compact group G acts as a measure preserving transformation group on a Lebesgue space X, then there is a naturally induced unitary representation of G on L2(X), and one can study the action on X by means of this representation. The situation in which the representation has discrete spectrum (i.e., is the direct sum of finite dimensional representations) and the action is ergodic was examined by von Neumann and Halmos when G is the integers or the real line [7], and by Mackey for general non-abelian G [10].

A new characterization of the Haagerup property by actions on infinite measure spaces

2020 ◽  
pp. 1-20
Author(s):  
THIEBOUT DELABIE ◽  
PAUL JOLISSAINT ◽  
ALEXANDRE ZUMBRUNNEN
Keyword(s):  
Finite Measure ◽  
Invariant Mean ◽  
Locally Compact ◽  
Haagerup Property ◽  
Infinite Measure ◽  
Measurable Functions ◽  
Measure Spaces ◽  
Definition Of

The aim of the article is to provide a characterization of the Haagerup property for locally compact, second countable groups in terms of actions on $\unicode[STIX]{x1D70E}$ -finite measure spaces. It is inspired by the very first definition of amenability, namely the existence of an invariant mean on the algebra of essentially bounded, measurable functions on the group.

Actions of a Locally Compact Group with Zero

1971 ◽  
Vol 23 (3) ◽  
pp. 413-420 ◽  
Author(s):  
T. H. McH. Hanson
Keyword(s):  
Topological Group ◽  
Compact Group ◽  
Topological Semigroup ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Definition Of ◽  
Isolated Point ◽  
Locally Compact Topological Group

In [2] we find the definition of a locally compact group with zero as a locally compact Hausdorff topological semigroup, S, which contains a non-isolated point, 0, such that G = S – {0} is a group. Hofmann shows in [2] that 0 is indeed a zero for S, G is a locally compact topological group, and the unit, 1, of G is the unit of S. We are to study actions of S and G on spaces, and the reader is referred to [4] for the terminology of actions.If X is a space (all are assumed Hausdorff) and A ⊂ X, A* denotes the closure of A. If {xρ} is a net in X, we say limρxρ = ∞ in X if {xρ} has no subnet which converges in X.

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