A local quantization principle for inclusions of tracial von Neumann algebras

TL;DR

This paper explores a local quantization principle for inclusions of tracial von Neumann algebras, demonstrating how elements can be approximated or averaged using projections within subalgebras, with specific results for type II_1 factors.

Contribution

It extends the local quantization principle to inclusions of tracial von Neumann algebras, providing explicit constructions and averaging results for type II_1 factors.

Findings

Existence of partitions approximating elements within epsilon in 2-norm.

Explicit averaging formulas for elements in type II_1 factors with index 2.

Construction of unitaries achieving element averaging in subalgebras.

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 $\mathcal{N}\subseteq \mathcal{M}$ is an inclusion of type $\rm II_{1}$ factors with $[\mathcal{M}:\mathcal{N}]=2$, then for any $x_{1},\ldots, x_{m}\in \mathcal{M}$, 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 $u\in \mathcal{N}$ such that \[\frac{1}{n}\sum_{i=1}^nu^{*i}x_j u^i=\tau(x_j)1, \quad 1\leq j\leq m.\]