Partial actions of groups and actions of inverse semigroups

Abstract:

Given a group $G$, we construct, in a canonical way, an inverse semigroup $\mathcal {S}(G)$ associated to $G$. The actions of $\mathcal {S}(G)$ 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 $\mathcal {S}(G)$ have the same representation theory. We show that $\mathcal S(G)$ 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 $Z/4 Z$ and $Z/2 Z \oplus Z/2 Z$, are not isomorphic.