Representation of atomic operators and extension problems

The notion of an atomic operator between spaces of measurable functions was introduced in 2002 in a paper by Drakhlin, Ponosov and Stepanov in order to provide a reasonable generalization of local operators useful for applications. It has been shown that, roughly speaking, atomic operators amount to compositions of local operators with shifts. A natural problem is then when a continuous-in-measure atomic operator can be represented as a composition of a Nemytskiǐ (composition) operator generated by a Carathéodory function, and a shift operator. In this paper we will show that the answer to this question is inherently related to the possibility of extending an atomic operator with continuity from a space of functions measurable with respect to some σ-algebra to a larger space of functions measurable with respect to a larger σ-algebra, as well as to the possibility of extending any σ-homomorphism from a smaller-measure algebra to a σ-homomorphism on a larger-measure algebra. We characterize precisely the condition on the respective σ-algebras which provides such possibilities and induces the positive answer to the above representation problem.