Bi-Free Entropy with Respect to a Completely Positive Map
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.