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Banach algebras on semigroups and on their compactifications. (English) Zbl 1192.43001

Mem. Am. Math. Soc. 966, v, 165 p. (2010).
Let \(S\) be a (discrete) semigroup. In the memoir under review the authors study the structure of the semigroup Banach algebra \(l^1(S)\) and its second dual \(l^1(S)''\). The algebra \(l^1(S)\) is taken with the convolution product \(*\) and the second dual \(l^1(S)''\) is taken with respect to the first and second Arens products, \(\square\) and \(\lozenge\); these second dual algebras are identified with the Banach algebras \(M(\beta S, \square)\) and \(M(\beta S, \lozenge)\), which are, hrespectively, the right and left topological semigroups of measures defined on \(\beta S\), the Stone-Čech compactification of \(S\).
The authors determine exactly when a semigroup algebra \(l^1(S)\) is amenable, so answering an open question, and discuss its amenability constant. It is proved that \(S\) is finite whenever \(M(\beta S)\) is amenable and it is discussed when \(M(\beta S)\) is weakly amenable. The authors 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. Also, they discuss left-invariant means on \(S\) as elements of the space \(M(\beta S)\) and the radical of the algebras \(l^1(\beta S)\) and \(M(\beta S)\). The memoir concludes with a list of selected open problems.

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
43-02 Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis
43A20 \(L^1\)-algebras on groups, semigroups, etc.
43A10 Measure algebras on groups, semigroups, etc.
46J10 Banach algebras of continuous functions, function algebras
46J45 Radical Banach algebras
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