Operator-valued free entropy and modular frames (Q819532)

We introduce the operator-valued relative free entropy \(\chi^*_{\mathcal B}(X_1,X_2,\dots,X_n:{\mathcal B})\) of a family of self-adjoint random variables \(X_1,X_2,\dots,X_n\) in a \({\mathcal B}\)-valued noncommutative probability space \(({\mathcal A},E_{\mathcal B},{\mathcal B})\). This notion extends D. Voiculescu's relative free entropy \(\Phi^*\) 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 \(\delta^*_{{\mathcal B},\tau}\).