On the paving size of a subfactor

TL;DR

This paper establishes bounds on the paving size for finite index subfactors of II$_1$ factors, providing a quantitative measure of how well the conditional expectation can be approximated by projections within the subfactor.

Contribution

It introduces explicit bounds on the paving size for subfactors with finite Jones index, advancing the understanding of approximation properties in operator algebras.

Findings

Bound on the number of projections needed for paving in terms of index and epsilon

Existence of partitions approximating the conditional expectation within epsilon

Introduction of invariants called 'paving size' for subfactors

Abstract

Given an inclusion of II$_1$ factors $N\subset M$ with finite Jones index, $[M:N]<\infty$, we prove that for any $F\subset M$ finite and $\varepsilon >0$, there exists a partition of $1$ with $r\leq \lceil 16\varepsilon^{-2}\rceil$ $\cdot \lceil 4 [M:N]\varepsilon^{-2}\rceil$ projections $p_1, ..., p_r\in N$ such that $\|\sum_{i=1}^r p_ixp_i - E_{N'\cap M}(x)\|\leq \varepsilon \|x-E_{N'\cap M}(x)\|$, $\forall x\in F$ (where $\lceil \beta \rceil$ denotes the least integer $\geq \beta$). We consider a series of related invariants for $N\subset M$, generically called {\it paving size}.