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.
"On nuclearity of the $C^*$-algebra of an inverse semigroup." Publ. Mat. 64 (2) 499 - 511, 2020. https://doi.org/10.5565/PUBLMAT6422005