A spectral characterization of Riesz homomorphisms between complex Riesz algebras

Let A be a complex Riesz algebra with a positive identity e. We show that a (not necessary linear) functional ϕ:A→C is a unital Riesz homomorphism if and only if ϕa−ϕb∈σPea−Peb for all a,b∈A, where Pe denotes the order projection onto the center edd of A. Then, as an application, we prove that unital Riesz homomorphisms, local unital Riesz homomorphisms, and 2-local Riesz homomorphisms between complex Riesz algebras with positive identities coincide.