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(2 ⁿ uоy) = h(2 ⁿ u) о h(y), h(3 ⁿ uо y) = h(3 ⁿ 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.