Operator-valued Free Entropy and Modular Frames

Abstract

We introduce the operator-valued relative free entropy $\chi_\mb^{\ast}(X_1,X_2,\cdots,X_n:\mb)$ of a family of self-adjoint random variables $X_1,X_2,\cdots,X_n$ in a $\mb$-valued noncommutative probability space $(\ma,\emb,\mb)$. This notion extends D. Voiculescu's relative free entropy $\Phi^{\ast}$ which defined in a tracial W*-noncommutative probability space to a more general context. The free entropy of a semicircular variable with conditional expectation covariance can be computed by using the modular frames and then we point out the relation between the free entropy of a semicircular variable and the index of a conditional expectation. At last, we obtain an estimate of the free entropy dimension $\delta^\ast_{\mb,\tau}$.