Strong Morita Equivalence of Inverse Semigroups

Abstract

We introduce strong Morita equivalence for inverse semigroups. This notion encompasses Mark Lawson's concept of enlargement. Strongly Morita equivalent inverse semigroups have Morita equivalent universal groupoids in the sense of Paterson and hence strongly Morita equivalent universal and reduced $C^{\ast }$-algebras. As a consequence we obtain a new proof of a result of Khoshkam and Skandalis showing that the $C^{\ast }$-algebra of an $F$-inverse semigroup is strongly Morita equivalent to a cross product of a commutative $C^{\ast }$-algebra by a group.

References in corpus (2)

• Etale groupoids and their quantales
• Graph inverse semigroups, groupoids and their C*-algebras

Cited by in corpus (12)

• A categorical invariant of flow equivalence of shifts
• Semigroup and Category-Theoretic Approaches to Partial Symmetry
• Functoriality of groupoid quantales. I
• $F$-inverse monoids as algebraic structures in enriched signature
• Morita equivalence of pseudogroups
• Equivalent groupoids have Morita equivalent Steinberg algebras
• Functoriality of groupoid quantales. II
• Generalized heaps, inverse semigroups and Morita equivalence
• Morita equivalence of semigroups with local units
• The principal bundles over an inverse semigroup
• Characterizations of Morita equivalent inverse semigroups
• Morita equivalence of inverse semigroups