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Covariant representations of actions of inverse semigroups: a new approach to the reduced and essential crossed-product C*-algebras

  • Astrid an Huef
  • Ilija Tolich
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Abstract

We consider an action of an inverse semigroup on a C*-algebra A and use it to construct a groupoid of germs with unit space the spectrum of A. Motivated by the representation theory of C*-algebras of groupoids, we construct a concrete family of covariant representations for the action. We use this family to give new definitions of the reduced and essential crossed product C*-algebras that avoid, respectively, passing to the double commutant and local multiplier algebra of A. Our reduced crossed product is isomorphic to the one defined by Exel, Buss and Meyer, and when the inverse semigroup is quasi-countable our essential crossed product is isomorphic to the one defined by Kwaśniewski and Meyer.

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Cite

BibTeXRIS
BibTeX
@misc{huef2026covariant,
  title = {{Covariant representations of actions of inverse semigroups: a new approach to the reduced and essential crossed-product C*-algebras}},
  author = {Astrid an Huef and Ilija Tolich},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25694},
  eprint = {2609.25694},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25694},
}
RIS
TY  - UNPB
TI  - Covariant representations of actions of inverse semigroups: a new approach to the reduced and essential crossed-product C*-algebras
AU  - Astrid an Huef
AU  - Ilija Tolich
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.25694
UR  - https://arxiv.org/abs/2609.25694
AB  - We consider an action of an inverse semigroup on a C*-algebra A and use it to construct a groupoid of germs with unit space the spectrum of A. Motivated by the representation theory of C*-algebras of groupoids, we construct a concrete family of covariant representations for the action. We use this family to give new definitions of the reduced and essential crossed product C*-algebras that avoid, respectively, passing to the double commutant and local multiplier algebra of A. Our reduced crossed product is isomorphic to the one defined by Exel, Buss and Meyer, and when the inverse semigroup is quasi-countable our essential crossed product is isomorphic to the one defined by Kwaśniewski and Meyer.
ER  -