Massoud Amini and Reza Rezavand: Module nuclearity and module injectivity of $C^{*}$-modules, p.357-366

Abstract:

We define module nuclearity and module injectivity for $C^{*}$-algebras which are $C^*$-module over another $C^*$-algebra with compatible actions and extend Connes-Haagerup result to this context by showing that module nuclearity is equivalent to module amenability. We also solve the module version of an open problem of Alan L.T. Paterson, by showing that the $C^*$-algebra of an inverse semigroup $S$ is module nuclear over the $C^*$-algebra of its idempotents if and only if $S$ is amenable.

Key Words: Operator algebras, operator amenability, module operator amenability, inverse semigroup, Fourier algebra.

2000 Mathematics Subject Classification: Primary: 43A07;
Secondary: 46L07.

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