Generalized notions of amenability.
type: Item
instance of: scholarly article
title: Generalized notions of amenability.
zbMATH Open document ID: 1045.46029
DOI: 10.1016/S0022-1236(03)00214-3
published in: Journal of Functional Analysis
publication date: 2004-03-29T00:00:00Z
review text: Let \(A\) be a Banach algebra, and let \(E\) be a Banach \(A\)-bimodule. A (bounded) derivation \(D : A \to E\) is said to be inner if there is \(x \in E\) such that \(D = \text{ad}_x\), where \[ \text{ad}_x a = a \cdot x - x \cdot a \qquad (a \in A). \] In his seminal memoir [Cohomology in Banach algebras, Mem. Am. Math. Soc. 127 (1972; Zbl 0256.18014)], \textit{B. E. Johnson} defined a Banach algebra to be amenable if, for each Banach \(A\)-bimodule \(E\), every derivation \(D : A \to E^\ast\) is inner. The choice of terminology was motivated by the following theorem: a locally compact group \(G\) is amenable if and only if its group algebra \(L^1(G)\) is amenable. Suppose now that \(A\) is a Banach algebra with the following property: for each Banach \(A\)-bimodule \(E\) and for each derivation \(D : A \to E^\ast\), there is a --- not necessarily bounded --- net \(( x_\alpha )_\alpha\) in \(E^\ast\) such that \(D a = \lim_\alpha \text{ad}_{x_\alpha} a\) for each \(a \in A\). The authors of the paper under review call such a Banach algebra approximately amenable. It is clear that every amenable Banach algebra is approximately amenable. However, as the authors show, the converse is false. In contrast, a locally compact group \(G\) is amenable if and only if \(L^1(G)\) is approximately amenable so that, for group algebras, the notions of amenability and approximate amenability coincide. The authors systematically investigate the hereditary properties of approximate amenability, and they establish a characterization of approximate amenability through an appropriate variant of approximate diagonals. In general, the theory of approximately amenable Banach algebras parallels that of amenable Banach algebras, but the results tend to be weaker and not as aesthetically pleasing. Besides approximate amenability, the authors discuss various other Banach algebraic properties related to (and mostly weaker than) amenability. The paper concludes with a list of open questions. Th…
reviewed by: Volker Runde
Mathematics Subject Classification ID: 46H20
Mathematics Subject Classification ID: 43A20
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 47B47
zbMATH DE Number: 2062736
zbMATH Keywords: amenability
zbMATH Keywords: approximate amenability
author: F. Ghahramani
author: R. J. Loy
MaRDI profile type: Publication
cites work: On the norm of an inner derivation of a C*-algebra
cites work: Q3215824
cites work: First order cohomology of Banach semigroup algebras
cites work: A Simple C*-Algebra Generated by Two Finite-Order Unitaries
cites work: The Exact Cardinality of the Set of Invariant Means on a Group
cites work: The Structure of Amenable Banach Algebras
cites work: Banach algebras and automatic continuity
cites work: The Amenability of Measure Algebras
cites work: Approximate identities and factorization in Banach modules
cites work: Isometric Isomorphisms Between the Second Conjugate Algebras of Group Algebras
cites work: Weak amenability of certain classes of Banach algebras without bounded approximate identities
cites work: Isometric Isomorphisms Between Banach Algebras Related to Locally Compact Groups
cites work: Amenability and weak amenability of second conjugate Banach algebras
cites work: Derivations on Group Algebras
cites work: Amenability of Lipschitz algebras
cites work: All nuclear C*-algebras are amenable
cites work: Q4510743
cites work: Approximate Diagonals and Cohomology of Certain Annihilator Banach Algebras
cites work: Non-Amenability of the Fourier Algebra of a Compact Group
cites work: Derivations and automorphisms of operator algebras. II
cites work: The Exact Cardinality of the Set of Topological Left Invariant Means on an Amenable Locally Compact Group
cites work: Identities in tensor products of Banach algebras
cites work: Q4040376
cites work: Q4520727
cites work: Lectures on amenability
cites work: Nilpotent ideals in a class of Banach algebras
full work available at URL: s0022-1236(03)00214-3
OpenAlex ID: W4213265382
description: scientific article; zbMATH DE number 2062736