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Inverse semigroup expansions and their actions on \(C^{\ast}\)-algebras. (English) Zbl 1298.46053

Summary: In this work, we give a presentation of the prefix expansion \(\mathbf {Pr}(G)\) of an inverse semigroup \(G\) as recently introduced by M. V. Lawson, S. W. Margolis and B. Steinberg [J. Aust. Math. Soc. 80, No. 2, 205–228 (2006; Zbl 1101.20030)] which is similar to the universal inverse semigroup defined by the second named author in the case where \(G\) is a group. The inverse semigroup \(\mathbf{Pr}(G)\) classifies the partial actions of \(G\) on spaces. We extend this result and prove that Fell bundles over \(G\) correspond bijectively to saturated Fell bundles over \(\mathbf{Pr}(G)\). In particular, this shows that twisted partial actions of \(G\) (on \(C^{*}\)-algebras) correspond to twisted (global) actions of \(\mathbf{Pr}(G)\). Furthermore, we show that this correspondence preserves \(C^{*}\)-algebra crossed products.

MSC:

46L55 Noncommutative dynamical systems
46L05 General theory of \(C^*\)-algebras
20M18 Inverse semigroups
20M30 Representation of semigroups; actions of semigroups on sets

Citations:

Zbl 1101.20030
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Full Text: arXiv Euclid

References:

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