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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2004, Volume 245, Pages 218–227
(Mi tm187)
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Poisson Algebra Homomorphisms and Poisson Brackets
Chun-Gil Park Department of Mathematics, Chungnam National University
Abstract:
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.
Received in October 2003
Citation:
Chun-Gil Park, “Poisson Algebra Homomorphisms and Poisson Brackets”, Selected topics of $p$-adic mathematical physics and analysis, Collected papers. Dedicated to the 80th birthday of academician Vasilii Sergeevich Vladimirov, Trudy Mat. Inst. Steklova, 245, Nauka, MAIK «Nauka/Inteperiodika», M., 2004, 218–227; Proc. Steklov Inst. Math., 245 (2004), 205–214
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https://www.mathnet.ru/eng/tm187 https://www.mathnet.ru/eng/tm/v245/p218
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Abstract page: | 394 | Full-text PDF : | 69 | References: | 30 |
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