Compressible subalgebras in II${}_1$ factors

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 $\text{Cal}{U}_0\subset \text{Cal}U(M)$ such that $\mathrm{\parallel }\frac{1}{\mathrm{\mid }\text{Cal}{U}_0\mathrm{\mid }}{\sum }_{u\in \text{Cal}{U}_0}ux{u}^{\ast }-E_{1\otimes {\mathbb{M}}_K(\mathbb{C})}(x)\mathrm{\parallel }\le \epsilon$, $\mathrm{\forall }K\ge 1$, $\mathrm{\forall }x\in (Q\otimes {\mathbb{M}}_K(\mathbb{C}){)}_1$. 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 $M\subset \text{Cal}M$ that are ergodic but not AFD-ergodic, even after stabilizing by $\text{Cal}B({\mathrm{\ell }}^2\mathbb{N})$.