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In this paper we introduce the notion of module character amenable Banach algebras and show that they possess module character virtual (approximate) diagonals. As a basic example, we show that for an inverse semigroup S with the set of idempotents E, the semigroup algebra ℓ1(S) is module character amenable as an ℓ1(E)-module if only if S is amenable.
In this paper we define the module topological center of the second dual of a Banach algebra which is a Banach -module with compatible actions on another Banach algebra . We calculate the module topological center of ℓ1(S)**, as an ℓ1(E)-module, for an inverse semigroup S with an upward directed set of idempotents E. We also...
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