Compressible subalgebras in II$_1$ factors
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914103627350016 |
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| author | Popa, Sorin |
| author_facet | Popa, Sorin |
| contents | Given a II$_1$ factor $M$, a W$^*$-subalgebra $Q\subset M$ is {\it compressible} if for any $\varepsilon>0$ there exists a finite set of unitary elements $\Cal U_0\subset \Cal U(M)$ such that $\| \frac{1}{|\Cal U_0|}\sum_{u\in \Cal U_0} uxu^* -E_{1\otimes \Bbb M_K(\Bbb C)}(x)\|\leq \varepsilon$, $\forall K\geq 1$, $\forall x\in (Q\otimes \Bbb M_K(\Bbb C))_1$. Any W$^*$-subalgebra $Q$ in a II$_1$ factor $M$ which admits a diffuse W$^*$-algebra $Q_0\subset M$ that's free independent to $Q$, is compressible in $M$. We prove that if $Q\subset M$ is compressible, then $_NL^2M_Q$ contains a copy of the coarse $N-Q$ bimodule for any AFD subalgebra $N\subset M$. We use this result to provide examples of inclusions of factors $M\subset \Cal M$ that are ergodic but not AFD-ergodic, even after stabilizing by $\Cal B(\ell^2\Bbb N)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17076 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Compressible subalgebras in II$_1$ factors Popa, Sorin Operator Algebras 46L10, 46L36 Given a II$_1$ factor $M$, a W$^*$-subalgebra $Q\subset M$ is {\it compressible} if for any $\varepsilon>0$ there exists a finite set of unitary elements $\Cal U_0\subset \Cal U(M)$ such that $\| \frac{1}{|\Cal U_0|}\sum_{u\in \Cal U_0} uxu^* -E_{1\otimes \Bbb M_K(\Bbb C)}(x)\|\leq \varepsilon$, $\forall K\geq 1$, $\forall x\in (Q\otimes \Bbb M_K(\Bbb C))_1$. Any W$^*$-subalgebra $Q$ in a II$_1$ factor $M$ which admits a diffuse W$^*$-algebra $Q_0\subset M$ that's free independent to $Q$, is compressible in $M$. We prove that if $Q\subset M$ is compressible, then $_NL^2M_Q$ contains a copy of the coarse $N-Q$ bimodule for any AFD subalgebra $N\subset M$. We use this result to provide examples of inclusions of factors $M\subset \Cal M$ that are ergodic but not AFD-ergodic, even after stabilizing by $\Cal B(\ell^2\Bbb N)$. |
| title | Compressible subalgebras in II$_1$ factors |
| topic | Operator Algebras 46L10, 46L36 |
| url | https://arxiv.org/abs/2510.17076 |