Partial actions of groups and actions of inverse semigroups

Abstract

Given a group G G , we construct, in a canonical way, an inverse semigroup S ( G ) \mathcal {S}(G) associated to G G . The actions of S ( G ) \mathcal {S}(G) are shown to be in one-to-one correspondence with the partial actions of G G , both in the case of actions on a set, and that of actions as operators on a Hilbert space. In other words, G G and S ( G ) \mathcal {S}(G) have the same representation theory. We show that S ( G ) \mathcal S(G) governs the subsemigroup of all closed linear subspaces of a G G -graded C ∗ {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 ∗ { C}^* -algebra of a discrete group is introduced. While the usual group C ∗ { C}^* -algebra of finite commutative groups forgets everything but the order of the group, we show that the partial group C ∗ { C}^* -algebra of the two commutative groups of order four, namely Z / 4 Z Z/4 Z and Z / 2 Z ⊕ Z / 2 Z Z/2 Z \oplus Z/2 Z , are not isomorphic.