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Additivity of Jordan$ \sp \ast$-maps on $ AW\sp \ast$-algebras


Author: Jôsuke Hakeda
Journal: Proc. Amer. Math. Soc. 96 (1986), 413-420
MSC: Primary 46L10
MathSciNet review: 822431
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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 [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0822431-0
Keywords: Jordan product, AW*-algebra, $ {C^*}$-algebra, projections lattice, additivity
Article copyright: © Copyright 1986 American Mathematical Society