An estimate of free entropy and applications

We obtain an estimate of free entropy of generators in a type ${II}_1$-factor $\text{mc}M$ which has a subfactor $\text{mc}N$ of finite index with a subalgebra $\text{mc}P=\text{mc}{P}_1\vee \text{mc}{P}_2\subset \text{mc}N$ where $\text{mc}{P}_1=\text{mc}{R}_1^{\mathrm{\prime }}\cap \text{mc}P$, $\text{mc}{P}_2=\text{mc}{R}_2^{\mathrm{\prime }}\cap \text{mc}P$ are diffuse, $\text{mc}{R}_1,\text{mc}{R}_2\subset \text{mc}P$ are mutually commuting hyperfinite subfactors, and an abelian subalgebra $\text{mc}A\subset \text{mc}N$ such that the correspondence ${}_{\text{mc}}P{L}^2(\text{mc}N,\tau {)}_{\text{mc}}A$ is $\text{mc}M$-weakly contained in a subcorrespondence ${}_{\text{mc}}P{H}_{\text{mc}}A$ of ${}_{\text{mc}}P{L}^2(\text{mc}M,\tau {)}_{\text{mc}}A$, generated by $v$ vectors. The (modified) free entropy dimension of any generating set of $\text{mc}M$ is $\le 2r+2v+4$, 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.