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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2412.13751 |
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| _version_ | 1866914410452221952 |
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| author | Austin, Tim |
| author_facet | Austin, Tim |
| contents | Ergodic theory includes several notions of entropy for probability-preserving actions of countable groups. These include Kolmogorov--Sinai entropy based on Følner sequences for amenable groups, entropy defined using a random ordering of the group, and Bowen's sofic entropy for sofic groups.
In this work we pursue these notions across an analogy between ergodic theory and representation theory. We arrive at new quantities associated to unitary representations of groups and representations of other C*-algebras. Our main results show that these new quantities can often be evaluated as Fuglede--Kadison determinants. The resulting determinantal formulas offer various non-commutative generalizations of Szegő's limit theorem for Toeplitz determinants. They make contact with Arveson's theory of subdiagonal subalgebras, and also with some entropy formulas in the ergodic theory of actions by automorphisms of compact Abelian groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13751 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Entropy and determinants for unitary representations Austin, Tim Operator Algebras Dynamical Systems Functional Analysis Probability Spectral Theory Primary: 43A35, 43A65, 46L30, 22D25, Secondary: 28D20, 28D15, 46K50, 46L51, 46L52 Ergodic theory includes several notions of entropy for probability-preserving actions of countable groups. These include Kolmogorov--Sinai entropy based on Følner sequences for amenable groups, entropy defined using a random ordering of the group, and Bowen's sofic entropy for sofic groups. In this work we pursue these notions across an analogy between ergodic theory and representation theory. We arrive at new quantities associated to unitary representations of groups and representations of other C*-algebras. Our main results show that these new quantities can often be evaluated as Fuglede--Kadison determinants. The resulting determinantal formulas offer various non-commutative generalizations of Szegő's limit theorem for Toeplitz determinants. They make contact with Arveson's theory of subdiagonal subalgebras, and also with some entropy formulas in the ergodic theory of actions by automorphisms of compact Abelian groups. |
| title | Entropy and determinants for unitary representations |
| topic | Operator Algebras Dynamical Systems Functional Analysis Probability Spectral Theory Primary: 43A35, 43A65, 46L30, 22D25, Secondary: 28D20, 28D15, 46K50, 46L51, 46L52 |
| url | https://arxiv.org/abs/2412.13751 |