Banach algebras on semigroups and on their compactifications

Dales, H.; Lau, A.; Strauss, D.

doi:10.1090/S0065-9266-10-00595-8

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Banach algebras on semigroups and on their compactifications

About this Title

H. G. Dales, Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom, A. T.-M. Lau, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada and D. Strauss, Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom

Publication: Memoirs of the American Mathematical Society
Publication Year: 2010; Volume 205, Number 966
ISBNs: 978-0-8218-4775-6 (print); 978-1-4704-0580-9 (online)
DOI: https://doi.org/10.1090/S0065-9266-10-00595-8
Published electronically: January 25, 2010
Keywords: Banach algebras, second duals, dual Banach algebras, Arens products, Arens regular, topological centre, strongly Arens irregular, introverted subspace, amenable, weakly amenable, approximately amenable, amenability constant, ultrafilter, Stone–Čech compactification, invariant mean, semigroup, Rees semigroup, semigroup algebra, Munn algebra, semi-character, character, cancellative semigroup, radical, minimal ideal, idempotent, locally compact group, group algebra, measure algebra, bounded approximate identity, diagonal
MSC: Primary 43A10, 43A20; Secondary 46J10

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1. Introduction
2. Banach algebras and their second duals
3. Semigroups
4. Semigroup algebras
5. Stone–Čech compactifications
6. The semigroup $(\beta S, \Box )$
7. Second duals of semigroup algebras
8. Related spaces and compactifications
9. Amenability for semigroups
10. Amenability of semigroup algebras
11. Amenability and weak amenability for certain Banach algebras
12. Topological centres
13. Open problems

Abstract

Let $S$ be a (discrete) semigroup, and let $\ell ^{\,1}( S )$ be the Banach algebra which is the semigroup algebra of $S$. We shall study the structure of this Banach algebra and of its second dual.

We shall 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.

The second dual of $\ell ^{\,1}( S )$ is the Banach algebra $M(\beta S)$ of measures on the Stone–Čech compactification $\beta S$ of $S$, where $M(\beta S)$ and $\beta S$ are taken with the first Arens product $\Box$. We shall show that $S$ is finite whenever $M(\beta S)$ is amenable, and we shall discuss when $M(\beta S)$ is weakly amenable. We shall show that the second dual of $L^1(G)$, for $G$ a locally compact group, is weakly amenable if and only if $G$ is finite.

We shall also discuss left-invariant means on $S$ as elements of the space $M(\beta S)$, and determine their supports.

We shall show that, for each weakly cancellative and nearly right cancellative semigroup $S$, the topological centre of $M(\beta S)$ is just $\ell ^{\,1}(S)$, and so $\ell ^{\,1}(S)$ is strongly Arens irregular; indeed, we shall considerably strengthen this result by showing that, for such semigroups $S$, there are two-element subsets of $\beta S \setminus S$ that are determining for the topological centre; for more general semigroups $S$, there are finite subsets of $\beta S \setminus S$ with this property.

We have partial results on the radical of the algebras $\ell ^{\,1}(\beta S )$ and $M(\beta S)$.

We shall also discuss analogous results for related spaces such as $WAP(S)$ and $LUC(G)$.

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Banach algebras on semigroups and on their compactifications

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memo_has_moved_text();Banach algebras on semigroups and on their compactifications

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Banach algebras on semigroups and on their compactifications