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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 $AW^ \ast$-algebras
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by Jôsuke Hakeda PDF
Proc. Amer. Math. Soc. 96 (1986), 413-420 Request permission


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.
  • Sterling K. Berberian, Baer *-rings, Die Grundlehren der mathematischen Wissenschaften, Band 195, Springer-Verlag, New York-Berlin, 1972. MR 0429975, DOI 10.1007/978-3-642-15071-5
  • 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  • J. Hakeda and K. Saitô, Additivity of $*$ -semigroup isomorphisms among AW* -algebras, unpublished.
  • Richard V. Kadison, Isometries of operator algebras, Ann. of Math. (2) 54 (1951), 325–338. MR 43392, DOI 10.2307/1969534
  • Irving Kaplansky, Rings of operators, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0244778
  • S. Strǎtilǎ and L. Zsidó, Lectures on von Neumann algebras, Abacus Press, Tunbridge Wells, 1979.
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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 96 (1986), 413-420
  • MSC: Primary 46L10
  • DOI:
  • MathSciNet review: 822431