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Banach algebra structure and amenability of a class of matrix algebras with applications. (English) Zbl 0927.46027

Let \({\mathcal A}\) be any unital Banach algebra, let \(I\) and \(J\) be any index sets and let \(P= (P_{ij})\) be a \(J\times I\)-matrix over \({\mathcal A}\) with \(\| P\|_\infty= \sup\{\| P_{ij}\|: i\in I, j\in J\}<\infty\). With these notations, the \(\ell^1\)-Munn matrix algebra \({\mathcal L}{\mathcal M}({\mathcal A}, P)\) over \({\mathcal P}\) with sandwich matrix \(P\) is the Banach algebra of all \(I\times J\)-matrices \(A= (A_{ij})\) with \(A_{ij}\in{\mathcal A}\) and \(\| A\|_1= \sum_{i\in I,j\in J}\| A_{ij}\|< \infty\), and a multiplication given by \(A\circ B= APB\). In the paper under review, the author relates the structure of \(\ell^1\)-Munn algebras \({\mathcal L}{\mathcal M}({\mathcal A}, P)\) to properties of the sandwich matrix \(P\) and the index sets \(I\), \(J\). In this direction he proves the following results:
a) \({\mathcal L}{\mathcal M}({\mathcal A},P)\) has a bounded approximate identity iff \(I\), \(J\) are finite and \(P\) is invertible.
b) \({\mathcal L}{\mathcal M}({\mathcal A}, P)\) is amenable iff \({\mathcal A}\) is amenable, \(I\), \(J\) are finite and \(P\) is invertible.
c) \({\mathcal L}{\mathcal M}({\mathcal A}, P)\) has a bounded approximate identity and \({\mathcal A}\) is semisimple iff \(I\), \(J\) are finite and \({\mathcal L}{\mathcal M}({\mathcal A}, P)\) is semisimple.
The results are applied to the study of the amenability of semigroup algebras \(\ell^1(S)\), and as a main result the following one is shown: Let \(S\) be any regular semigroup with only a finite number of idempotents. Then \(\ell^1(S)\) is amenable iff every maximal subgroup of \(S\) is amenable and if \(\ell^1(T)\) is semisimple for every principal factor \(T\) of \(S\). This generalizes the well-known result for inverse semigroups. The paper ends with some examples and counterexamples for \(\ell^1(S)\).
Reviewer: H.Junek (Potsdam)

MSC:

46H20 Structure, classification of topological algebras
46H05 General theory of topological algebras
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References:

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