On the paving size of a subfactor

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 $\epsilon >0$, there exists a partition of $1$ with $r\le \lceil 16{\epsilon }^{-2}\rceil$ $\cdot \lceil 4[M:N]{\epsilon }^{-2}\rceil$ projections $p_1,...,p_r\in N$ such that $\Vert {\sum }_{i=1}^r{p}_{i}x{p}_i-E_{N^{\prime }\cap M}(x)\Vert \le \epsilon \Vert x-E_{N^{\prime }\cap M}(x)\Vert$, $\forall x\in F$ (where $\lceil \beta \rceil$ denotes the least integer $\ge \beta$). We consider a series of related invariants for $N\subset M$, generically called {\it paving size}.

9 pages, paper is dedicated to the memory of Vaughan Jones and Mihai Pimsner; minor corrections throughout the paper, final version, to appear in Pure and Applied Math Quarterly volume dedicated to Vaughan Jones; a few more typos found during galley proofs