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 ${\rm II}1$ von Neumann algebra and let $\mathcal{N}\subseteq \mathcal{M}$ be a type ${\rm II}_1$ von Neumann subalgebra. Let $x_1,\ldots, x_m \in \mathcal{M}$ and $ \epsilon> 0$. Then there exists a partition of 1 with projections $p{1}, \ldots, p_{n}$ 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}<\epsilon,\quad 1\leq j\leq m.] In particular, if $\N\subseteq \M$ is an inclusion of type $\rm II{1}$ factors with $[\M:\N]=2$, then for any $x_{1},\ldots, x_{m}\in \M$, there exists a partition of 1 with projections $p_{1}, \ldots, p_{n}$ in $\mathcal{N}$ such that [\sum_{i=1}^n p_ix_jp_i=\tau(x_j)1, \quad 1\leq j\leq m.] Equivalently, there exists a unitary operator $u\in \N$ such that [\frac{1}{n}\sum_{i=1}^nu^{*i}x_j u^i=\tau(x_j)1, \quad 1\leq j\leq m.]