On nuclearity of the \(C^*\)-algebra of an inverse semigroup. (English) Zbl 1460.46049

Summary: 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 [Groupoids, inverse semigroups, and their operator algebras. Boston, MA: Birkhäuser (1999; Zbl 0913.22001)].


46L35 Classifications of \(C^*\)-algebras
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
43A07 Means on groups, semigroups, etc.; amenable groups


Zbl 0913.22001
Full Text: DOI Euclid


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