Let $S$ be a (discrete) semigroup, and let $\ell^{\,1}(S)$ be the Banach algebra which is the semigroup algebra of $S$. The authors study the structure of this Banach algebra and of its second dual. The authors determine exactly when $\ell^{\,1}(S)$ is amenable as a Banach algebra, and shall discuss its amenability constant, showing that there are 'forbidden values' for this constant. Table of Contents: Introduction; Banach algebras and their second duals; Semigroups; Semigroup algebras; Stone-?ech compactifications; The semigroup $(\beta S, \Box)$; Second duals of semigroup algebras; Related spaces and compactifications; Amenability for semigroups; Amenability of semigroup algebras; Amenability and weak amenability for certain Banach algebras; Topological centres; Open problems; Bibliography; Index of terms; Index of symbols. (MEMO/205/966)

background

Weak Amenability of Discrete Semigroup Algebras

Banach Algebras on Semigroups and on Their Compactifications

We shall study the biflatness of the convolution algebra l 1(S) for a semigroup S. We show that for any semigroup S such that l 1(S) is biflat the canonical partial ordering on the idempotents must be uniformly locally finite. We use this to characterize the biflatness of l 1(S) for an inverse semigroup S.

background or methods

Biflatness of semigroup algebras

In recent years, there have been several studies of various `approximate' versions of the key notion of amenability, which is defined for all Banach algebras; these studies began with work of Ghahramani and Loy in 2004 The present memoir continues such work: we shall define various notions of approximate amenability, and we shall discuss and extend the known background, which considers the relationships between different versions of approximate amenability There are a number of open questions on these relationships; these will be considered In Chapter 1, we shall give all the relevant definitions and a number of basic results, partly surveying existing work; we shall concentrate on the case of Banach function algebras In Chapter 2, we shall discuss these properties for the semigroup algebra `1(S) of a semigroup S In the case where S has only finitely many idempotents, `1(S) is approximately amenable if and only if it is amenable In Chapter 3, we shall consider the class of weighted semigroup algebras of the form `1(N^; !), where ! : Z ! [1; 1) is an arbitrary function We shall determine necessary and sufficient conditions on ! for these Banach sequence algebras to have each of the various approximate amenability properties that interest us In this way we shall illuminate the implications between these properties In Chapter 4, we shall discuss Segal algebras on T and on R It is a conjecture that every proper Segal algebra on T fails to be approximately amenable; we shall establish this conjecture for a wide class of Segal algebras

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Approximate amenability of semigroup algebras and Segal algebras

We characterize cyclic and weakly amenable ' 1 -Munn algebras. In the special case of Rees matrix semigroups, we obtain a new proof of the following result due to Blackmore: The semigroup algebra of every Rees matrix semigroup is weakly amenable. Char- acterizations of Connes-amenable ' 1 -Munn algebras with square sandwich matrix and semigroup algebras of Rees matrix semigroups are also provided.

http://bims.iranjournals.ir/article_59_959a86b771414779ca0c9ec5a16e71a9.pdf

First order cohomology of l1-munn algebras and certain semigroup algebras

Let S be a Rees matrix semigroup. We show that l 1(S) is (2k + 1)-weakly amenable for k ∈ ℤ+.

Weak Amenability of Discrete Semigroup Algebras

Citations

Banach Algebras on Semigroups and on Their Compactifications

Biflatness of semigroup algebras

Approximate amenability of semigroup algebras and Segal algebras

First order cohomology of l1-munn algebras and certain semigroup algebras

On n -weak amenability of Rees semigroup algebras

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