Poisson Algebra Homomorphisms and Poisson Brackets

It is shown that every almost linear mapping $h:\mathcal A\rightarrow\mathcal B$ of a unital Poisson Banach algebra $\mathcal A$ to a unital Poisson Banach algebra $\mathcal B$ is a Poisson algebra homomorphism when $h(x y) = h(x) h(y)$ for all $x, y \in\mathcal A$, and that every almost linear almost multiplicative mapping $h:\mathcal A \rightarrow \mathcal B$ is a Poisson algebra homomorphism when $h(2x) = h(2x)$ or $h(3x) = 3h(x)$ for all $x\in\mathcal A$. Here, the numbers $2$ and $3$ depend on the functional equations given in the almost linear almost multiplicative mappings. We prove that every almost Poisson bracket $B:\mathcal A\times\mathcal A\rightarrow\mathcal A$ on a Banach algebra $\mathcal A$ is a Poisson bracket when $B(2x,z) = B(x,2z) = 2B(x,z)$ or $B(3x,z) = B(x,3z) = 3B(x,z)$ for all $x,z\in\mathcal A$. Here, the numbers $2$ and $3$ depend on the functional equations given in the almost Poisson brackets.