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Motivated by a number of important examples of C*-algebras generated by an inverse semigroup of partial isometries, we look to use the algebraic theory of inverse semigroups to describe the structure of the C*-algebras they generate.
We begin by investigating the notion of amenability for inverse semigroups. We give evidence that the weak containment property, which is equivalent to amenability when restricted to groups, is an appropriate notion of amenability for inverse semigroups. We characterize the weak containment property for a number of examples: E-unitary inverse semigroups; bisimple inverse ω-semigroups; graph inverse semigroups, which we prove always have weak containment and the inverse semigroups associated with quasi-lattice ordered groups.
Next we study an important representation in semigroup theory, the Munn representation, which has had relatively little attention in the study of C*-algebras of inverse semigroups. Finally, using a partial action built from the Munn representation, we prove that the C*-algebra of a strongly 0-E-unitary inverse semigroup S can by realized as a partial crossed product of a group acting on the semilattice of idempotents of S.
