The non-commutative legendre-fenchel transform

摘要

We use the theory of matrix convex sets of Effros-Winkler to introduce a non-commutative version of convex functions, and we show how the Legendre-Fenchel transform generalizes to this situation based on the analogue of the classical theorem that any closed convex function is the supremum over all affine functions dominated by the convex function.