Compressible subalgebras in II${}_1$ factors

Abstract

Given a II${}_1$ factor $M$, a W${}^{\ast }$-subalgebra $Q\subset M$ is {\it compressible} if for any $\epsilon >0$ there exists a finite set of unitary elements $\Cal U_0\subset \Cal U(M)$ such that $\| \frac{1}{|\Cal U_0|}\sum_{u\in \Cal U_0} uxu^* -E_{1\otimes \Bbb M_K(\Bbb C)}(x)\|\leq \varepsilon$, $\forall K\ge 1$, $\forall x\in (Q\otimes {\mathbb{M}}_K(\mathbb{C}){)}_1$. Any W${}^{\ast }$-subalgebra $Q$ in a II${}_1$ factor $M$ which admits a diffuse W${}^{\ast }$-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 ${}_N{L}^2{M}_Q$ 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 $\Cal B(\ell^2\Bbb N)$.

21 pages; final version, 24 pages, many small corrections and additions, including a dedication to the memory of Huzihiro Araki, to appear in Comm. Math. Physics