On the involutive Banach algebra associated to topologically free dynamical systems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918394379370496 |
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| author | Roland, Tabaré |
| author_facet | Roland, Tabaré |
| contents | Given an action $G \curvearrowright X$ of a discrete and countable infinite group $G$ on a compact and Hausdorff space $X$, we regard $\ell^1(G\curvearrowright X)$ as the Banach *-algebra crossed product associated to the action. We characterize topological freeness of the action by showing that it is equivalent to every nontrivial closed ideal of $\ell^1(G\curvearrowright X)$ intersecting $C(X)$ nontrivially. Most surprisingly, we show that when $G$ is torsion-free and abelian, $\ell^1(G\curvearrowright X)$ can detect freeness of $G \curvearrowright X$: indeed, we show that $G\curvearrowright X$ is free if and only if every closed ideal of $\ell^1(G\curvearrowright X)$ is self-adjoint, a property that is automatic in $C^*$-algebras. We also show with an example that this result does not hold beyond the torsion-free abelian case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00108 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the involutive Banach algebra associated to topologically free dynamical systems Roland, Tabaré Functional Analysis Dynamical Systems Operator Algebras 47L65 (Primary) 37B05, 46H10 (Secondary) Given an action $G \curvearrowright X$ of a discrete and countable infinite group $G$ on a compact and Hausdorff space $X$, we regard $\ell^1(G\curvearrowright X)$ as the Banach *-algebra crossed product associated to the action. We characterize topological freeness of the action by showing that it is equivalent to every nontrivial closed ideal of $\ell^1(G\curvearrowright X)$ intersecting $C(X)$ nontrivially. Most surprisingly, we show that when $G$ is torsion-free and abelian, $\ell^1(G\curvearrowright X)$ can detect freeness of $G \curvearrowright X$: indeed, we show that $G\curvearrowright X$ is free if and only if every closed ideal of $\ell^1(G\curvearrowright X)$ is self-adjoint, a property that is automatic in $C^*$-algebras. We also show with an example that this result does not hold beyond the torsion-free abelian case. |
| title | On the involutive Banach algebra associated to topologically free dynamical systems |
| topic | Functional Analysis Dynamical Systems Operator Algebras 47L65 (Primary) 37B05, 46H10 (Secondary) |
| url | https://arxiv.org/abs/2505.00108 |