Samantha Butler
Abstract Conic quasi-linear maps are nonlinear operators from $$C_0(X)$$ 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, for $$E = C_b(Y)$$ E = C b ( Y ) a quasi-linear map is bounded iff it is continuous. $$E = {\mathbb {R}}$$ E = R gives quasi-integrals, which correspond to (deficient) topological measures—nonsubadditive set functions generalizing measures. Like image measures $$\mu \circ u^{-1}$$ μ ∘ u - 1 , (d-) image transformations move (deficient) topological measures from one space to another, generalizing $$u^{-1}$$ u - 1 . We give criteria for a (d-) image transformation to be $$u^{-1}$$ 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 $$E = C_0(Y), X, Y$$ E = C 0 ( Y ) , X , Y are locally compact. (Conic) quasi-homomorphisms behave like homomorphisms on singly generated subalgebras or cones. We show by construction 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. We show that one may approach a well-known result from Gelfand duality theory from the perspective of quasi-homomorphisms and image transformations. Any conic quasi-linear map or 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.