Coactions of compact groups on $M_n$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914847488212992 |
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| author | Kaliszewski, S. Landstad, Magnus B. Quigg, John |
| author_facet | Kaliszewski, S. Landstad, Magnus B. Quigg, John |
| contents | 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 $M_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16839 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coactions of compact groups on $M_n$ Kaliszewski, S. Landstad, Magnus B. Quigg, John Operator Algebras 46L05, 46L55 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 $M_n$. |
| title | Coactions of compact groups on $M_n$ |
| topic | Operator Algebras 46L05, 46L55 |
| url | https://arxiv.org/abs/2406.16839 |