Homomorphisms between Poisson JC*-Algebras

Abstract.

It is shown that every almost linear mapping $h:{\user1{\mathcal{A}}} \to {\user1{\mathcal{B}}}$ of a unital Poisson JC*-algebra ${\user1{\mathcal{A}}}$ to a unital Poisson JC*-algebra ${\user1{\mathcal{B}}}$ is a Poisson JC*-algebra homomorphism when h(2ⁿ uоy) = h(2ⁿ u) о h(y), h(3ⁿ uо y) = h(3ⁿ u) о h(y) or h(q ⁿ u о y) = h(q ⁿ u) о h(y) for all $y \in {\user1{\mathcal{A}}}$, all unitary elements $u \in {\user1{\mathcal{A}}}$ and n = 0, 1, 2, · · · , and that every almost linear almost multiplicative mapping $h:{\user1{\mathcal{A}}} \to {\user1{\mathcal{B}}}$ is a Poisson JC*-algebra homomorphism when h(2x) = 2h(x), h(3x) = 3h(x) or h(qx) = qh(x) for all $x \in {\user1{\mathcal{A}}}$. Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings or in the almost linear almost multiplicative mappings.

Moreover, we prove the Cauchy–Rassias stability of Poisson JC*-algebra homomorphisms in Poisson JC*-algebras.