On the topological ranks of Banach $^*$-algebras associated with groups of subexponential growth
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918320144384000 |
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| author | Flores, Felipe I. |
| author_facet | Flores, Felipe I. |
| contents | Let $G$ be a group of subexponential growth and $\mathscr C\overset{q}{\to}G$ a Fell bundle. We show that any Banach $^*$-algebra that sits between the associated $\ell^1$-algebra $\ell^1( G\,\vert\,\mathscr C)$ and its $C^*$-envelope has the same topological stable rank and real rank as $\ell^1( G\,\vert\,\mathscr C)$. We apply this result to compute the topological stable rank and real rank of various classes of symmetrized twisted $L^p$-crossed products and show that some twisted $L^p$-crossed products have topological stable rank 1. Our results are new even in the case of (untwisted) group algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15437 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the topological ranks of Banach $^*$-algebras associated with groups of subexponential growth Flores, Felipe I. Operator Algebras Functional Analysis Primary 46K05, Secondary 43A20, 43A15 Let $G$ be a group of subexponential growth and $\mathscr C\overset{q}{\to}G$ a Fell bundle. We show that any Banach $^*$-algebra that sits between the associated $\ell^1$-algebra $\ell^1( G\,\vert\,\mathscr C)$ and its $C^*$-envelope has the same topological stable rank and real rank as $\ell^1( G\,\vert\,\mathscr C)$. We apply this result to compute the topological stable rank and real rank of various classes of symmetrized twisted $L^p$-crossed products and show that some twisted $L^p$-crossed products have topological stable rank 1. Our results are new even in the case of (untwisted) group algebras. |
| title | On the topological ranks of Banach $^*$-algebras associated with groups of subexponential growth |
| topic | Operator Algebras Functional Analysis Primary 46K05, Secondary 43A20, 43A15 |
| url | https://arxiv.org/abs/2503.15437 |