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Partial actions of groups and actions of inverse semigroups. (English) Zbl 0910.46041
Summary: 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.

46L05 General theory of \(C^*\)-algebras
46L55 Noncommutative dynamical systems
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