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Proceedings of the American Mathematical Society

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

 

 

Jordan-morphisms in $ \ast $-algebras


Author: Klaus Thomsen
Journal: Proc. Amer. Math. Soc. 86 (1982), 283-286
MSC: Primary 46K05; Secondary 46L05
DOI: https://doi.org/10.1090/S0002-9939-1982-0667290-0
MathSciNet review: 667290
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Abstract: As a continuation of Størmer's work on Jordan-morphisms in $ C*$-algebras we consider Jordan-morphisms $ \varphi $ from $ *$-algebras $ \mathfrak{A}$ into the $ *$-algebra $ B(\mathcal{H})$, and assume that $ \varphi (\mathfrak{A})$ is again a $ *$-algebra. We then establish the existence of three mutually orthogonal central projections $ {P_i}$, $ i = 1,2,3$, in $ \varphi {\left( {} \right)^{\prime \prime }}$ such that $ {P_1} + {P_2} + {P_3} = I$ and $ \varphi ( \cdot ){P_1}$ is a morphism, $ \varphi ( \cdot ){P_2}$ is an antimorphism. $ {P_3}$ is the largest projection such that $ \varphi ( \cdot ){P_3}$ is a morphism, as well as an antimorphism.

Uniqueness is also shown. The theorem improves a result of Kadison and Størmer. Our proofs are self-contained.


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DOI: https://doi.org/10.1090/S0002-9939-1982-0667290-0
Article copyright: © Copyright 1982 American Mathematical Society