A local quantization principle for inclusions of tracial von Neumann algebras

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 ${\mathrm{I}\mathrm{I}}_1$ von Neumann algebra and let $\mathcal{N}\subseteq \mathcal{M}$ be a type ${\mathrm{I}\mathrm{I}}_1$ 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 $${\parallel \sum _{i=1}^n{p}_i(x_j-E_{{\mathcal{N}}^{\mathrm{\prime }}\cap \mathcal{M}}(x_j))p_i\parallel }_2<\epsilon ,1\le j\le m\mathrm{.}$$ 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 $$\sum _{i=1}^n{p}_i{x}_j{p}_i=\tau (x_j)1,1\le j\le m\mathrm{.}$$ Equivalently, there exists a unitary operator $u\in \mathcal{N}$ such that $$\frac{1}{n}\sum _{i=1}^n{u}^{\ast i}{x}_j{u}^i=\tau (x_j)1,1\le j\le m\mathrm{.}$$