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

Abstract

We show that the universal groupoid of an inverse semigroup $S$ is topologically (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 $C^*$-algebra $C^*(E_S)$ on the idempotents $E_S$ of $S$ are amenable. Thereby we solve some problems raised by A. L. T. Paterson.