Free entropy, as introduced by Voiculescu in the 1990s, has been an important construct in free probability and an essential tool in proving many results in operator algebras. The non-microstate version of free entropy, which is based on a conjugate variable system and a notion of free Fisher information, was generalized by Shlyakhtenko to allow for the incorporation of a completely positive map into the conjugate variable expressions. In this talk, we will examine this notion of free entropy with respect to a completely positive map, its applications, and the extension of this concept to the operator-valued bi-free setting.

This is joint work with G. Katsimpas. This work was supported by NSERC grant RGPIN-2017-05711.