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Topologically free actions and ideals in twisted Banach algebra crossed products

Part of: Linear spaces and algebras of operators Connections with other structures, applications Rings and algebras arising under various constructions

Published online by Cambridge University Press:  26 April 2024

Krzysztof Bardadyn Open the ORCID record for Krzysztof Bardadyn [Opens in a new window]  and
Bartosz Kwaśniewski Open the ORCID record for Bartosz Kwaśniewski [Opens in a new window]
Krzysztof Bardadyn
Affiliation:
Faculty of Mathematics, University of Białystok, ul. K. Ciołkowskiego 1M, 15-245 Białystok, Poland (k.bardadyn@uwb.edu.pl;b.kwasniewski@uwb.edu.pl)
Bartosz Kwaśniewski
Affiliation:
Faculty of Mathematics, University of Białystok, ul. K. Ciołkowskiego 1M, 15-245 Białystok, Poland (k.bardadyn@uwb.edu.pl;b.kwasniewski@uwb.edu.pl)
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Abstract

We generalize the influential $C^*$-algebraic results of Kawamura–Tomiyama and Archbold–Spielberg for crossed products of discrete groups actions to the realm of Banach algebras and twisted actions. We prove that topological freeness is equivalent to the intersection property for all reduced twisted Banach algebra crossed products coming from subgroups, and in the untwisted case to a generalized intersection property for a full $L^p$-operator algebra crossed product for any $p\in [1,\,\infty ]$. This gives efficient simplicity criteria for various Banach algebra crossed products. We also use it to identify the prime ideal space of some crossed products as the quasi-orbit space of the action. For amenable actions we prove that the full and reduced twisted $L^p$-operator algebras coincide.

Keywords

crossed producttopological freenesssimplicity

MSC classification

Primary: 47L10: Algebras of operators on Banach spaces and other topological linear spaces
Secondary: 16S35: Twisted and skew group rings, crossed products 54H15: Transformation groups and semigroups
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 31
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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