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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.


References [Enhancements On Off] (What's this?)

  • [1] N. Jacobson and C. Rickart, Jordan homomorphisms of rings, Trans. Amer. Math. Soc. 69 (1950), 479-502. MR 0038335 (12:387h)
  • [2] R. V. Kadison, Isometries of operator algebras, Ann. of Math. (2) 54 (1951), 325-338. MR 0043392 (13:256a)
  • [3] -, Transformation of states in operator theory and dynamics, Topology 3 (1965), 177-198. MR 0169073 (29:6328)
  • [4] E. Størmer, On the Jordan structure of $ {C^*}$-algebras, Trans. Amer. Math. Soc. 120 (1965), 438-447. MR 0185463 (32:2930)
  • [5] O. Bratteli and D. W. Robinson, Operator algebras and quantum statistical mechanics. I, Springer-Verlag, New York, 1979. MR 611508 (82k:82013)
  • [6] I. N. Herstein, Jordan homomorphisms, Trans. Amer. Math. Soc. 81 (1956), 331-341. MR 0076751 (17:938f)

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

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