# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Additivity of Jordan$^ \ast$-maps on $AW^ \ast$-algebrasHTML articles powered by AMS MathViewer

by Jôsuke Hakeda
Proc. Amer. Math. Soc. 96 (1986), 413-420 Request permission

## Abstract:

Let $M$ and $N$ be AW*-algebras and $\phi$ be a Jordan*-map from $M$ to $N$ which satisfies (1) $\phi (x \circ y) = \phi (x) \circ \phi (y)$ for all $x$ and $y$ in $M$, (2) $\phi ({x^*}) = \phi {(x)^*}$ for all $x \in M$, and (3) $\phi$ is bijective, where $x \circ y = (1/2)(xy + yx)$. If $M$ has no abelian direct summand and a Jordan*-map $\phi$ is uniformly continuous on every abelian ${C^*}$-subalgebra of $M$, then we can conclude that $\phi$ is additive. Moreover, $\phi$ is the sum of ${\phi _i}(i = 1,2,3,4)$ such that ${\phi _1}$ is a linear $*$-ring isomorphism, ${\phi _2}$ is a linear $*$-ring anti-isomorphism, ${\phi _3}$ is a conjugate linear $*$-ring anti-isomorphism and ${\phi _4}$ is a conjugate linear $*$-ring isomorphism.
References
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• Jôsuke Hakeda, Characterizations of properly infinite von Neumann algebras, Math. Japon. 31 (1986), no. 5, 707–710. MR 872793
• J. Hakeda and K. Saitô, Additivity of $*$ -semigroup isomorphisms among AW* -algebras, unpublished.
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