Geodesic
Poisson Algebra Homomorphisms and Poisson Brackets
Informatics and Automation, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 218-227.

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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. 
@article{TRSPY_2004_245_a21,
     author = {Chun-Gil Park},
     title = {Poisson {Algebra} {Homomorphisms} and {Poisson} {Brackets}},
     journal = {Informatics and Automation},
     pages = {218--227},
     publisher = {mathdoc},
     volume = {245},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2004_245_a21/}
}
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Chun-Gil Park. Poisson Algebra Homomorphisms and Poisson Brackets. Informatics and Automation, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 218-227. http://geodesic.mathdoc.fr/item/TRSPY_2004_245_a21/

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