On the singular abelian rank of ultraproduct II$_1$ factors
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911785810919424 |
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| author | Hiatt, Patrick Popa, Sorin |
| author_facet | Hiatt, Patrick Popa, Sorin |
| contents | We prove that, under the continuum hypothesis $\frak c=\aleph_1$, any ultraproduct II$_1$ factor $M= \prod_ω M_n$ of separable finite factors $M_n$ contains more than $\frak c$ many mutually disjoint singular MASAs, in other words the {\it singular abelian rank of} $M$, $\text{\rm r}(M)$, is larger than $ \frak c$. Moreover, if the strong continuum hypothesis $2^{\frak c}=\aleph_2$ is assumed, then ${\text{\rm r}}(M) = 2^{\frak c}$. More generally, these results hold true for any II$_1$ factor $M$ with unitary group of cardinality $\frak c$ that satisfies the bicommutant condition $(A_0'\cap M)'\cap M=M$, for all $A_0\subset M$ separable abelian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_18559 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the singular abelian rank of ultraproduct II$_1$ factors Hiatt, Patrick Popa, Sorin Operator Algebras 46L10, 46L36 We prove that, under the continuum hypothesis $\frak c=\aleph_1$, any ultraproduct II$_1$ factor $M= \prod_ω M_n$ of separable finite factors $M_n$ contains more than $\frak c$ many mutually disjoint singular MASAs, in other words the {\it singular abelian rank of} $M$, $\text{\rm r}(M)$, is larger than $ \frak c$. Moreover, if the strong continuum hypothesis $2^{\frak c}=\aleph_2$ is assumed, then ${\text{\rm r}}(M) = 2^{\frak c}$. More generally, these results hold true for any II$_1$ factor $M$ with unitary group of cardinality $\frak c$ that satisfies the bicommutant condition $(A_0'\cap M)'\cap M=M$, for all $A_0\subset M$ separable abelian. |
| title | On the singular abelian rank of ultraproduct II$_1$ factors |
| topic | Operator Algebras 46L10, 46L36 |
| url | https://arxiv.org/abs/2402.18559 |