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Super module amenability of inverse semigroup algebras. (English) Zbl 1277.46022

Summary: We compare the notions of super amenability and super module amenability of Banach algebras which are Banach modules over another Banach algebra with compatible actions. We find conditions for the two notions to be equivalent. In particular, we study arbitrary module actions of \(l^1(E_S)\) on \(l^1(S)\) for an inverse semigroup \(S\) with the set of idempotents \(E_S\) and show that under certain conditions, \(l^1(S)\) is super module amenable if and only if \(S\) is finite. We also study the super module amenability of \(l^1(S)^{\ast \ast }\) and module biprojectivity of \(l^1(S)\), for arbitrary actions.

MSC:

46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
43A20 \(L^1\)-algebras on groups, semigroups, etc.
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