2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73032Suppose $N \subset M$ is an inclusion of $II_1$-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 $δ_0(X) \geq δ_0(Y)$, where $δ_0$ denotes Voiculescu's microstates (modified) free entropy dimension. Moreover given $ε>0$ one has $δ_0(F) \geq δ_0(G) \geq ([M:N]^{-2} -ε) \cdot (δ_0(F) -1) + 1 - ε$ for certain generating sets $F$ for $N$ and $G$ for $M$.11 pagesOperator Algebras46L54; 46L35; 52C17text