Group von Neumann algebras, inner amenability, and unit groups of continuous rings
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916636745793536 |
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| author | Schneider, Friedrich Martin |
| author_facet | Schneider, Friedrich Martin |
| contents | We prove that, if a discrete group $G$ is not inner amenable, then the unit group of the ring of operators affiliated with the group von Neumann algebra of $G$ is non-amenable with respect to the topology generated by its rank metric. This provides examples of non-discrete irreducible, continuous rings (in von Neumann's sense) whose unit groups are non-amenable with regard to the rank topology. Our argument establishes and uses connections with Eymard--Greenleaf amenability of the action of the unitary group of a $\mathrm{II}_{1}$ factor on the associated space of projections of a fixed trace. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_03537 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Group von Neumann algebras, inner amenability, and unit groups of continuous rings Schneider, Friedrich Martin Operator Algebras Functional Analysis Group Theory 43A07, 22D25, 06C20, 16E50 We prove that, if a discrete group $G$ is not inner amenable, then the unit group of the ring of operators affiliated with the group von Neumann algebra of $G$ is non-amenable with respect to the topology generated by its rank metric. This provides examples of non-discrete irreducible, continuous rings (in von Neumann's sense) whose unit groups are non-amenable with regard to the rank topology. Our argument establishes and uses connections with Eymard--Greenleaf amenability of the action of the unitary group of a $\mathrm{II}_{1}$ factor on the associated space of projections of a fixed trace. |
| title | Group von Neumann algebras, inner amenability, and unit groups of continuous rings |
| topic | Operator Algebras Functional Analysis Group Theory 43A07, 22D25, 06C20, 16E50 |
| url | https://arxiv.org/abs/2211.03537 |