Back-and-forth equivalent group von Neumann algebras
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917621499166720 |
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| author | Goldbring, Isaac Harrison-Trainor, Matthew |
| author_facet | Goldbring, Isaac Harrison-Trainor, Matthew |
| contents | We prove that if $G$ and $H$ are $α$-back-and-forth equivalent groups (in the sense of computable structure theory) for some ordinal $α\geq ω$, then their group von Neumann algebras $L(G)$ and $L(H)$ are also $α$-back-and-forth equivalent. In particular, if $G$ and $H$ are $ω$-back-and-forth-equivalent groups, then $L(G)$ and $L(H)$ are elementarily equivalent; this is known to fail under the weaker hypothesis that $G$ and $H$ are merely elementarily equivalent. We extend this result to crossed product von Neumann algebras associated to Bernoulli actions of back-and-forth equivalent groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16181 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Back-and-forth equivalent group von Neumann algebras Goldbring, Isaac Harrison-Trainor, Matthew Logic Operator Algebras We prove that if $G$ and $H$ are $α$-back-and-forth equivalent groups (in the sense of computable structure theory) for some ordinal $α\geq ω$, then their group von Neumann algebras $L(G)$ and $L(H)$ are also $α$-back-and-forth equivalent. In particular, if $G$ and $H$ are $ω$-back-and-forth-equivalent groups, then $L(G)$ and $L(H)$ are elementarily equivalent; this is known to fail under the weaker hypothesis that $G$ and $H$ are merely elementarily equivalent. We extend this result to crossed product von Neumann algebras associated to Bernoulli actions of back-and-forth equivalent groups. |
| title | Back-and-forth equivalent group von Neumann algebras |
| topic | Logic Operator Algebras |
| url | https://arxiv.org/abs/2403.16181 |