Vaĭnberg--Brègman geometry and quasinonexpansive operators
Ryshard-Pavel Kostecki
TL;DR
This paper develops a unified Vaĭnberg--Brègman framework on reflexive Banach spaces and extends it to nonreflexive spaces via nonlinear embeddings, enabling a broad theory of entropic projections and quasinonexpansive operators. It introduces extended functionals D_{ell,Psi} that pull back convex geometry through embeddings, and builds categories of left/right projections (lCvx, bar rCvx) and strongly quasinonexpansive maps (LSQ/RSQ) with adjunctions and monads, facilitating compositional operator theory. The work provides extensive model realizations, including noncommutative L_p spaces, Mazur maps, Lozanovskii factorization, finite base-normed spaces, and preduals of W^*- and JBW-algebras, and establishes pythagorean-type equalities/inequalities, continuity and Lipschitz properties, and adaptation results. These results bridge norm-geometry with information-geometric structures, offering a robust toolkit for nonlinear convex analysis and operator theory in both classical and quantum settings, with potential applications to quantum information, base-normed state spaces, and noncommutative geometry.
Abstract
We obtain an array of new results in the theory of Vaĭnberg--Brègman relative entropies and quasinonexpansive operators on reflexive Banach spaces, and we also develop an extension of this theory to nonreflexive Banach spaces, providing a joint generalisation of the reflexive approach with the finite-dimensional information geometric approach. In the reflexive case, we study generalised pythagorean inequality, as well as norm-to-norm, uniform, and Lipschitz--Hölder continuity of entropic projections, proximal maps, and resolvents. We also provide a detailed account of the family of Vaĭnberg--Brègman geometries that are naturally encoding the geometric properties of the underlying Banach space norm. The extended theory belongs to the intersection of convex theoretic and homeomorphic approaches to nonlinear analysis. Its models are constructed using integration theory on order unit spaces. Our main focus is on geometry of, and nonlinear quasinonexpansive operators on, normal state spaces of JBW- and W*-algebras, modelled via nonlinear embeddings into reflexive rearrangement invariant spaces. For example, we analyse the extended Vaĭnberg--Brègman geometries induced by norm geometry of noncommutative and nonassociative L$_p$ spaces via Mazur embeddings, providing sufficient conditions for monoidal composability of extended quasinonexpansive operators, and computing the exponent parameters of Lipschitz--Hölder continuity of the corresponding extended entropic projections and resolvents. Other models feature the (commutative and noncommutative) Lozanovskiĭ factorisation map, generalised spin factors, finite dimensional base normed spaces, and convex spectral functions on unitarily invariant ideals of compact operators. We also discuss several categories of entropic projections and quasinonexpansive operators naturally appearing in this framework.