Weak*-Simplicity of Convolution Algebras on Discrete Groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929414539837440 |
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| author | White, Jared T. |
| author_facet | White, Jared T. |
| contents | We prove that, given a discrete group $G$, and $1 \leq p < \infty$, the algebra of $p$-convolution operators $CV_p(G)$ is weak*-simple, in the sense of having no non-trivial weak*-closed ideals, if and only if $G$ is an ICC group. This generalises the basic fact that $vN(G)$ is a factor if and only if $G$ is ICC. When $p=1$, $CV_p(G) = \ell^1(G)$. In this case we give a more detailed analysis of the weak*-closed ideals, showing that they can be described in terms of the weak*-closed ideals of $\ell^1(FC(G))$; when $FC(G)$ is finite, this leads to a classification of the weak*-closed ideals of $\ell^1(G)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_15570 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weak*-Simplicity of Convolution Algebras on Discrete Groups White, Jared T. Functional Analysis Operator Algebras 43A15, 43A20 (primary), 47L10, 46H10 (secondary) We prove that, given a discrete group $G$, and $1 \leq p < \infty$, the algebra of $p$-convolution operators $CV_p(G)$ is weak*-simple, in the sense of having no non-trivial weak*-closed ideals, if and only if $G$ is an ICC group. This generalises the basic fact that $vN(G)$ is a factor if and only if $G$ is ICC. When $p=1$, $CV_p(G) = \ell^1(G)$. In this case we give a more detailed analysis of the weak*-closed ideals, showing that they can be described in terms of the weak*-closed ideals of $\ell^1(FC(G))$; when $FC(G)$ is finite, this leads to a classification of the weak*-closed ideals of $\ell^1(G)$. |
| title | Weak*-Simplicity of Convolution Algebras on Discrete Groups |
| topic | Functional Analysis Operator Algebras 43A15, 43A20 (primary), 47L10, 46H10 (secondary) |
| url | https://arxiv.org/abs/2309.15570 |