Banach Journal of Mathematical Analysis

Partial actions of C-quantum groups

Franziska Kraken, Paula Quast, and Thomas Timmermann

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Abstract

Partial actions of groups on C-algebras and the closely related actions and coactions of Hopf algebras have received much attention in recent decades. They arise naturally as restrictions of their global counterparts to noninvariant subalgebras, and the ambient enveloping global (co)actions have proven useful for the study of associated crossed products. In this article, we introduce the partial coactions of C-bialgebras, focusing on C-quantum groups, and we prove the existence of an enveloping global coaction under mild technical assumptions. We also show that partial coactions of the function algebra of a discrete group correspond to partial actions on direct summands of a C-algebra, and we relate partial coactions of a compact or its dual discrete C-quantum group to partial coactions or partial actions of the dense Hopf subalgebra. As a fundamental example, we associate to every discrete C-quantum group a quantum Bernoulli shift.

Article information

Source
Banach J. Math. Anal., Volume 12, Number 4 (2018), 843-872.

Dates
Received: 23 June 2017
Accepted: 1 September 2017
First available in Project Euclid: 30 January 2018

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1517281422

Digital Object Identifier
doi:10.1215/17358787-2017-0056

Mathematical Reviews number (MathSciNet)
MR3858752

Zentralblatt MATH identifier
06946294

Subjects
Primary: 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]
Secondary: 16T20: Ring-theoretic aspects of quantum groups [See also 17B37, 20G42, 81R50]

Keywords
$C^{*}$-algebra partial action quantum group Hopf algebra globalization

Citation

Kraken, Franziska; Quast, Paula; Timmermann, Thomas. Partial actions of $C^{*}$ -quantum groups. Banach J. Math. Anal. 12 (2018), no. 4, 843--872. doi:10.1215/17358787-2017-0056. https://projecteuclid.org/euclid.bjma/1517281422


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