An estimate of free entropy and applications

Abstract

We obtain an estimate of free entropy of generators in a type ${II}_1$-factor $\mc{M}$ which has a subfactor $\mc{N}$ of finite index with a subalgebra $\mc{P}=\mc{P}_1\vee\mc{P}_2\subset\mc{N}$ where $\mc{P}_1=\mc{R}_1'\cap\mc{P}$, $\mc{P}_2=\mc{R}_2'\cap\mc{P}$ are diffuse, $\mc{R}_1,\mc{R}_2\subset\mc{P}$ are mutually commuting hyperfinite subfactors, and an abelian subalgebra $\mc{A}\subset\mc{N}$ such that the correspondence $_\mc{P}L^2(\mc{N},\tau)_\mc{A}$ is $\mc{M}$-weakly contained in a subcorrespondence $_\mc{P}H_\mc{A}$ of $_\mc{P}L^2(\mc{M},\tau)_\mc{A}$, generated by $v$ vectors. The (modified) free entropy dimension of any generating set of $\mc{M}$ is $\leq 2r+2v+4$, where $r$ is the integer part of the index. As a consequence, the interpolated free group subfactors of finite index do not have regular non-prime subfactors or regular diffuse hyperfinite subalgebras.