Some free entropy dimension inequalities for subfactors

Abstract

Suppose $N\subset M$ is an inclusion of $I{I}_1$-factors of finite index. If $N$ can be generated by a finite set of elements, then there exist finite generating sets $X$ for $N$ and $Y$ for $M$ such that ${\delta }_0(X)\ge {\delta }_0(Y)$, where ${\delta }_0$ denotes Voiculescu's microstates (modified) free entropy dimension. Moreover given $\epsilon >0$ one has ${\delta }_0(F)\ge {\delta }_0(G)\ge ([M:N{]}^{-2}-\epsilon )\cdot ({\delta }_0(F)-1)+1-\epsilon$ for certain generating sets $F$ for $N$ and $G$ for $M$.

11 pages