On the paving size of a subfactor

Abstract

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