It is shown that every almost linear mapping

of a unital Poisson Banach algebra

to a unital Poisson Banach algebra

is a Poisson algebra homomorphism when h(xy) = h(x)h(y) holds for all

, and that every almost linear almost multiplicative mapping

is a Poisson algebra homomorphism when h(qx) = qh(x) for all

. Here the number q is in the functional equation given in the almost linear almost multiplicative mapping. We prove that every almost Poisson bracket

on a Banach algebra

is a Poisson bracket when B(qx, z) = B(x, qz) = qB(x, z) for all

. Here the number q is in the functional equation given in the almost Poisson bracket.