Quasi-measures and image transformations on completely regular spaces

A map ρ :C(X)→C(Y) is called quasi-linear if it is linear on each singly generated subalgebra of C(X). We prove the representation theorem for positive quasi-linear functionals in terms of integration with respect to quasi-measures. We then consider positive quasi-linear maps in more generality and show that they can be described in terms of transformations of sets in the underlying spaces. We call these image transformations. It is shown how smoothness properties of the quasi-linear map are reflected in the corresponding image transformation and the simplifications that occur when the underlying spaces are compact.