Poisson Banach modules over a Poisson C*-algebra

Abstract

It is shown that every almost linear mapping h: A → B of a unital Poisson C*-algebra A to a unital Poisson C*-algebra B is a Poisson C*-algebra homomorphism when h(2nuy) = h(2nu)h(y) or h(3nuy) = h(3nu)h(y) for all y 2 A, all unitary elements u ∈ A and n = 0, 1, 2, and that every almost linear almost multiplicative mapping h: A → B is a Poisson C*-algebra homomorphism when h(2x) = 2h(x) or h(3x) = 3h(x) for all x ∈ A. Here the numbers 2, 3 depend on the functional equations given in the almost linear mappings or in the almost linear almost multiplicative mappings. We prove the Cauchy-Rassias stability of Poisson C*-algebra homomorphisms in unital Poisson C*-algebras, and of homomorphisms in Poisson Banach modules over a unital Poisson C*-algebra.