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 Memoirs of the American Mathematical Society
Memoirs of the American Mathematical Society
ISSN 1947-6221(e) ISSN 0065-9266(p)

     

Banach algebras on semigroups and on their compactifications

Author(s): H. G. Dales; A. T.-M. Lau; D. Strauss
Journal: Memoirs of the AMS 205 (2010), no. 966.
MSC (2000): Primary 43A10, 43A20; Secondary 46J10
Posted: January 25, 2010
MathSciNet review: 2650729
Retrieve article in: PDF

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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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Additional Information:

H. G. Dales
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
Email: garth@maths.leeds.ac.uk

A. T.-M. Lau
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Email: tlau@math.ualberta.ca

D. Strauss
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
Email: d.strauss@maths.leeds.ac.uk

DOI: 10.1090/S0065-9266-10-00595-8
PII: S 0065-9266(10)00595-8
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--\v {C}ech 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
Received by editor(s): September 8, 2006
Posted: January 25, 2010
Copyright of article: Copyright 2010, American Mathematical Society




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