Continuity and equivariant dimension
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917336440635392 |
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| author | Chirvasitu, Alexandru Passer, Benjamin |
| author_facet | Chirvasitu, Alexandru Passer, Benjamin |
| contents | We study the local-triviality dimensions of actions on $C^*$-algebras, which are invariants developed for noncommutative Borsuk-Ulam theory. While finiteness of the local-triviality dimensions is known to guarantee freeness of an action, we show that free actions need not have finite weak local-triviality dimension. Moreover, the local-triviality dimensions of a continuous field may be greater than those of its individual fibers, and the dimensions may fail to vary continuously across the fibers. However, in certain circumstances upper semicontinuity of the weak local-triviality dimension is guaranteed. We examine these results and counterexamples with a focus on noncommutative tori and noncommutative spheres, both in terms of computation and theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_06767 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Continuity and equivariant dimension Chirvasitu, Alexandru Passer, Benjamin Operator Algebras Algebraic Topology Functional Analysis Group Theory Quantum Algebra 46L55, 46L65, 46L80, 46L85, 46M20, 55R25, 55S40 We study the local-triviality dimensions of actions on $C^*$-algebras, which are invariants developed for noncommutative Borsuk-Ulam theory. While finiteness of the local-triviality dimensions is known to guarantee freeness of an action, we show that free actions need not have finite weak local-triviality dimension. Moreover, the local-triviality dimensions of a continuous field may be greater than those of its individual fibers, and the dimensions may fail to vary continuously across the fibers. However, in certain circumstances upper semicontinuity of the weak local-triviality dimension is guaranteed. We examine these results and counterexamples with a focus on noncommutative tori and noncommutative spheres, both in terms of computation and theory. |
| title | Continuity and equivariant dimension |
| topic | Operator Algebras Algebraic Topology Functional Analysis Group Theory Quantum Algebra 46L55, 46L65, 46L80, 46L85, 46M20, 55R25, 55S40 |
| url | https://arxiv.org/abs/2403.06767 |