Quasi-linear maps and image transformations
S. V. Butler
TL;DR
The paper develops a unified framework linking nonlinear conic/quasi-linear maps, (d-) image transformations, and deficient/topological measures on locally compact spaces. It shows that image transformations correspond to conic quasi-homomorphisms and that bounded quasi-linear maps admit integral representations against a family of topological measures, with a fundamental decomposition $\theta = H \circ \Psi$ where $H$ is an algebra homomorphism. A central achievement is the 1-1 correspondence between (d-) image transformations and (conic) quasi-homomorphisms, extended to continuous $k$-proper maps, which clarifies how quasi-integrals and their nonlinear adjoints move between spaces. The results generalize image-measure concepts to deficient topological measures, extend previously compact-space theories to locally compact settings, and lay groundwork for nonlinear Markov-Feller-type operators arising as adjoints.
Abstract
Conic quasi-linear maps are nonlinear operators from $C_0(X)$ to a normed linear space $E$ which preserve nonnegative linear combinations on positive cones generated by single functions; quasi-linear maps are linear on singly generated subalgebras. While nonlinear, a quasi-linear map is bounded iff it is continuous. $E = \mathbb{R}$ gives quasi-integrals, which correspond to (deficient) topological measures - nonsubadditive set functions generalizing measures. Like image measures $μ\circ u^{-1}$, (d-) image transformations move (deficient) topological measures from one space to another, generalizing $u^{-1}$. We give criteria for a (d-) image transformation to be $u^{-1}$ for some proper continuous function. We study the interrelationships between (conic) quasi-linear maps, quasi-integrals, (deficient) topological measures and (d-) image transformations when $E = C_0(Y), X, Y$ are locally compact. (Conic) quasi-homomorphisms behave like homomorphisms on singly generated subalgebras or cones. We show that (conic) quasi-homomorphisms are in 1-1 correspondence with (d-) image transformations and with certain continuous proper functions. We give criteria for a (conic) quasi-linear map to be a (conic) quasi-homomorphism, and for the latter to be an algebra homomorphism. Any conic quasi-linear map or bounded quasi-linear map is a composition of an algebra homomorphism with the basic quasi-linear map, and we give criteria for the latter to be linear. We study the adjoints of (d-) image transformations and (conic) quasi-linear maps; for (conic) quasi-homomorphisms they give Markov-Feller operators with nonlinear duals.