A local quantization principle for inclusions of tracial von Neumann algebras

Abstract

We study the local quantization principle (after Sorin Popa~\cite{popa 94} and \cite{popa 95}) of inclusions of tracial von Neumann algebras. Let $(\mathcal{M},\tau )$ be a type ${\mathrm{II}}_1$ von Neumann algebra and let $\mathcal{N}\subseteq \mathcal{M}$ be a type ${\mathrm{II}}_1$ von Neumann subalgebra. Let $x_1,\dots ,x_m\in \mathcal{M}$ and $\epsilon >0$. Then there exists a partition of 1 with projections $p_1,\dots ,p_n$ in $\mathcal{N}$ such that \[\left\|\sum_{i=1}^n p_{i}\left(x_j-E_{\mathcal{N}'\cap \mathcal{M}}(x_j)\right)p_{i}\right\|_{2}<ε,\quad 1\leq j\leq m.\] In particular, if $\mathcal{N}\subseteq \mathcal{M}$ is an inclusion of type $\mathrm{I}{\mathrm{I}}_1$ factors with $[\mathcal{M}:\mathcal{N}]=2$, then for any $x_1,\dots ,x_m\in \mathcal{M}$, there exists a partition of 1 with projections $p_1,\dots ,p_n$ in $\mathcal{N}$ such that \[\sum_{i=1}^n p_ix_jp_i=τ(x_j)1, \quad 1\leq j\leq m.\] Equivalently, there exists a unitary operator $u\in \mathcal{N}$ such that \[\frac{1}{n}\sum_{i=1}^nu^{*i}x_j u^i=τ(x_j)1, \quad 1\leq j\leq m.\]