Towards postquantum Va\u{\i}nberg$-$Br\`{e}gman relative entropies

Abstract

We develop a new approach to construction of the Va\u{\i}nberg$-$Br\`{e}gman relative entropies over nonreflexive Banach spaces, based on nonlinear embeddings into reflexive Banach spaces. We apply it to derive some new families of Va\u{\i}nberg$-$Br\`{e}gman relative entropies over some radially compact base normed spaces in spectral duality, and to establish their basic properties. In particular, we prove (left and right) generalised pythagorean theorem and norm-to-norm continuity of the left entropic projections for a family of Va\u{\i}nberg$-$Br\`{e}gman relative entropies induced on preduals of any W$^*$-algebras (resp., semifinite JBW-algebras) using Mazur maps into noncommutative (resp., nonassociative) $L_p$ spaces, and on preduals of semifinite W$^*$-algebras using Kaczmarz maps into noncommutative Orlicz spaces. We also prove left generalised pythagorean theorem for a family of Va\u{\i}nberg$-$Br\`{e}gman relative entropies over preduals of generalised spin factors. Additionally, we characterise strict convexity, Gateaux differentiability, Riesz$-$Radon$-$Shmul'yan property, and reflexivity of the Morse$-$Transue$-$Nakano and Orlicz norms on noncommutative Orlicz spaces, establish Lipschitz$-$H\"{o}lder continuity of the nonassociative Mazur map on positive parts of unit balls, and introduce a new class of $L_p$ spaces over spectrally dual order unit spaces.