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COACTIONS OF COMPACT GROUPS ON Mn

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  31 October 2025

S. KALISZEWSKI ,
MAGNUS B. LANDSTAD Open the ORCID record for MAGNUS B. LANDSTAD [Opens in a new window]  and
JOHN QUIGG
S. KALISZEWSKI
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA e-mail: kaliszewski@asu.edu
MAGNUS B. LANDSTAD
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway e-mail: magnus.landstad@ntnu.no
JOHN QUIGG*
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
*
e-mail: quigg@asu.edu
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Abstract

We prove that every coaction of a compact group on a finite-dimensional $C^*$-algebra is associated with a Fell bundle. Every coaction of a compact group on a matrix algebra is implemented by a unitary operator. A coaction of a compact group on $M_n$ is inner if and only if its fixed-point algebra has an abelian $C^*$-subalgebra of dimension n. Investigating the existence of effective ergodic coactions on $M_n$ reveals that $\operatorname {SO}(3)$ has them, while $\operatorname {SU}(2)$ does not. We give explicit examples of the two smallest finite nonabelian groups having effective ergodic coactions on $M_n$.

Keywords

coactioninnerergodiccocycle

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L55: Noncommutative dynamical systems

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , First View , pp. 1 - 18
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This research is part of the EU Staff Exchange project 101086394 ‘Operator Algebras That One Can See’. It was partially supported by the University of Warsaw Thematic Research Programme ‘Quantum Symmetries’.

We dedicate this paper to the memory of our friend and colleague Iain Raeburn

Communicated by Aidan Sims

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