Partial actions of groups and actions of inverse semigroups

TL;DR

This paper constructs an inverse semigroup from a group G to study partial actions, showing their equivalence and exploring applications to G-graded C*-algebras and partial group C*-algebras, revealing new structural insights.

Contribution

It introduces the inverse semigroup S(G) associated with G, establishing a one-to-one correspondence with partial actions and analyzing their role in C*-algebra structures.

Findings

S(G) captures all partial actions of G

S(G) governs subsemigroups of G-graded C*-algebras

Partial group C*-algebras distinguish between certain finite groups

Abstract

Given a group G, we construct, in a canonical way, an inverse semigroup S(G) associated to G. The actions of S(G) are shown to be in one-to-one correspondence with the partial actions of G, both in the case of actions on a set, and that of actions as operators on a Hilbert space. In other words, G and S(G) have the same representation theory. We show that S(G) governs the subsemigroup of all closed linear subspaces of a G-graded C*-algebra, generated by the grading subspaces. In the special case of finite groups, the maximum number of such subspaces is computed. A ``partial'' version of the group C*-algebra of a discrete group is introduced. While the usual group C*-algebra of finite commutative groups forgets everything but the order of the group, we show that the partial group C*-algebra of the two commutative groups of order four, namely Z/4 and Z/2+Z/2, are not isomorphic.