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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.14153 |
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| _version_ | 1866916581628444672 |
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| author | Arulseelan, Jananan Goldbring, Isaac Hart, Bradd Sinclair, Thomas |
| author_facet | Arulseelan, Jananan Goldbring, Isaac Hart, Bradd Sinclair, Thomas |
| contents | We introduce the notion of a totally ($K$-) bounded element of a W*-probability space $(M, φ)$ and, borrowing ideas of Kadison, give an intrinsic characterization of the $^*$-subalgebra $M_{tb}$ of totally bounded elements. Namely, we show that $M_{tb}$ is the unique strongly dense $^*$-subalgebra $M_0$ of totally bounded elements of $M$ for which the collection of totally $1$-bounded elements of $M_0$ is complete with respect to the $\|\cdot\|_φ^\#$-norm and for which $M_0$ is closed under all operators $h_a(\log(Δ))$ for $a \in \mathbb{N}$, where $Δ$ is the modular operator and $h_a(t):=1/\cosh(t-a)$ (see Theorem 4.3). As an application, we combine this characterization with Rieffel and Van Daele's bounded approach to modular theory to arrive at a new language and axiomatization of W*-probability spaces as metric structures. Previous work of Dabrowski had axiomatized W*-probability spaces using a smeared version of multiplication, but the subalgebra $M_{tb}$ allows us to give an axiomatization in terms of the original algebra operations. Finally, we prove the (non-)axiomatizability of several classes of W*-probability spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14153 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Totally Bounded Elements in W*-probability Spaces Arulseelan, Jananan Goldbring, Isaac Hart, Bradd Sinclair, Thomas Operator Algebras Logic 46L10 (Primary) 46L53, 03C66 (Secondary) We introduce the notion of a totally ($K$-) bounded element of a W*-probability space $(M, φ)$ and, borrowing ideas of Kadison, give an intrinsic characterization of the $^*$-subalgebra $M_{tb}$ of totally bounded elements. Namely, we show that $M_{tb}$ is the unique strongly dense $^*$-subalgebra $M_0$ of totally bounded elements of $M$ for which the collection of totally $1$-bounded elements of $M_0$ is complete with respect to the $\|\cdot\|_φ^\#$-norm and for which $M_0$ is closed under all operators $h_a(\log(Δ))$ for $a \in \mathbb{N}$, where $Δ$ is the modular operator and $h_a(t):=1/\cosh(t-a)$ (see Theorem 4.3). As an application, we combine this characterization with Rieffel and Van Daele's bounded approach to modular theory to arrive at a new language and axiomatization of W*-probability spaces as metric structures. Previous work of Dabrowski had axiomatized W*-probability spaces using a smeared version of multiplication, but the subalgebra $M_{tb}$ allows us to give an axiomatization in terms of the original algebra operations. Finally, we prove the (non-)axiomatizability of several classes of W*-probability spaces. |
| title | Totally Bounded Elements in W*-probability Spaces |
| topic | Operator Algebras Logic 46L10 (Primary) 46L53, 03C66 (Secondary) |
| url | https://arxiv.org/abs/2501.14153 |