Generalised Brègman relative entropies: a brief introduction

We present some basic elements of the theory of generalised Brègman relative entropies over nonreflexive Banach spaces. Using nonlinear embeddings of Banach spaces together with the Euler–Legendre functions, this approach unifies two former approaches to Brègman relative entropy: one based on reflexive Banach spaces, another based on differential geometry. This construction allows to extend Brègman relative entropies, and related geometric and operator structures, to arbitrary-dimensional state spaces of probability, quantum, and postquantum theory. We give several examples, not considered previously in the literature.