## (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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### References:

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