Coactions of compact groups on $M_n$

S. Kaliszewski; Magnus B. Landstad; John Quigg

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arXiv · 2406.16839

Coactions of compact groups on $M_n$

S. Kaliszewski·Magnus B. Landstad·John Quigg

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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 $M_n$.

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S. Kaliszewski, Magnus B. Landstad, John Quigg. 2024-06-24. Coactions of compact groups on $M_n$. https://arxiv.org/abs/2406.16839

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