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Semicrossed products of C*-algebras and their C*-envelopes

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Abstract

Let C be a C*-algebra and a: C → C a unital *-endomorphism. There is a natural way to construct operator algebras, called semicrossed products, using a convolution induced by the action of a on C. We show that the C*-envelope of a semicrossed product is (a full corner of) a crossed product. As a consequence, we get that when a is *-injective, the semicrossed products are completely isometrically isomorphic and share the same C*-envelope, the crossed product \({C_\infty }\;{ \rtimes _{{\alpha _\infty }}}\mathbb{Z}\). We show that minimality of the dynamical system (C, a) is equivalent to non-existence of non-trivial Fourier invariant ideals in the C*-envelope. We get sharper results for commutative dynamical systems.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Newcastle University, Newcastle Upon Tyne, NE1 7RU, UK

    Evgenios T. A. Kakariadis

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  1. Evgenios T. A. Kakariadis
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Correspondence to Evgenios T. A. Kakariadis.

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The author was supported by the “Herakleitus II” program (co-financed by the European Social Fund, the European Union and Greece.)

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Kakariadis, E.T.A. Semicrossed products of C*-algebras and their C*-envelopes. JAMA 129, 1–31 (2016). https://doi.org/10.1007/s11854-016-0026-8

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  • DOI: https://doi.org/10.1007/s11854-016-0026-8

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