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(Super) module amenability, module topological centre and semigroup algebras. (English) Zbl 1207.46039

Summary: For a Banach algebra \(\mathcal{A}\) which is also an \(\mathfrak{A}\)-bimodule, we study relations between module amenability of closed subalgebras of \(\mathcal{A}''\), which contain \(\mathcal{A}\), and module Arens regularity of \(\mathcal{A}\) and the role of the module topological centre in module amenability of \(\mathcal{A}''\). Then we apply these results to \(\mathcal{A}=l^{1}(S)\) and \(\mathfrak{A}=l^{1}(E)\) for an inverse semigroup \(S\) with subsemigroup \(E\) of idempotents. We also show that \(l^{1}(S)\) is module amenable (equivalently, \(S\) is amenable) if and only if an appropriate group homomorphic image of \(S\), the discrete group \(\frac{S}{\approx}\), is amenable. Moreover, we define super module amenability and show that \(l ^{1}(S)\) is super module amenable if and only if \(\frac{S}{\approx}\) is finite.

MSC:

46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
43A07 Means on groups, semigroups, etc.; amenable groups
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