On amenability constants of Fourier algebras: new bounds and new examples
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| Format: | Preprint |
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2025
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| _version_ | 1866908935386038272 |
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| author | Choi, Yemon Ghandehari, Mahya |
| author_facet | Choi, Yemon Ghandehari, Mahya |
| contents | Let $G$ be a locally compact group. If $G$ is finite then the amenability constant of its Fourier algebra, denoted by ${\rm AM}({\rm A}(G))$, admits an explicit formula [Johnson, JLMS 1994]; if $G$ is infinite then no such formula for ${\rm AM}({\rm A}(G))$ is known, although lower and upper bounds were established by Runde [PAMS 2006]. Using non-abelian Fourier analysis, we obtain a sharper upper bound for ${\rm AM}({\rm A}(G))$ when $G$ is discrete. Combining this with previous work of the first author [Choi, IMRN 2023], we exhibit new examples of discrete groups and compact groups where ${\rm AM}({\rm A}(G))$ can be calculated explicitly; previously this was only known for groups that are products of finite groups with ``degenerate'' cases. Our new examples also provide additional evidence to support the conjecture that Runde's lower bound for the amenability constant is in fact an equality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_05243 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On amenability constants of Fourier algebras: new bounds and new examples Choi, Yemon Ghandehari, Mahya Functional Analysis Operator Algebras 43A30, 46H20 (primary), 20C15, 22D10 (secondary) Let $G$ be a locally compact group. If $G$ is finite then the amenability constant of its Fourier algebra, denoted by ${\rm AM}({\rm A}(G))$, admits an explicit formula [Johnson, JLMS 1994]; if $G$ is infinite then no such formula for ${\rm AM}({\rm A}(G))$ is known, although lower and upper bounds were established by Runde [PAMS 2006]. Using non-abelian Fourier analysis, we obtain a sharper upper bound for ${\rm AM}({\rm A}(G))$ when $G$ is discrete. Combining this with previous work of the first author [Choi, IMRN 2023], we exhibit new examples of discrete groups and compact groups where ${\rm AM}({\rm A}(G))$ can be calculated explicitly; previously this was only known for groups that are products of finite groups with ``degenerate'' cases. Our new examples also provide additional evidence to support the conjecture that Runde's lower bound for the amenability constant is in fact an equality. |
| title | On amenability constants of Fourier algebras: new bounds and new examples |
| topic | Functional Analysis Operator Algebras 43A30, 46H20 (primary), 20C15, 22D10 (secondary) |
| url | https://arxiv.org/abs/2507.05243 |