On nuclearity of the $C^*$-algebra of an inverse semigroup

Abstract

: We show that the universal groupoid of an inverse semigroup S is topo- logically (measurewise) amenable if and only if S is hyperfinite and all members of a family of subsemigroups of S indexed by the spectrum of the commutative C ∗ -al-gebra C ∗ ( E S ) on the idempotents E S of S are amenable. Thereby we solve some problems raised by A. L. T. Paterson.