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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A result concerning additive functions in Hermitian Banach $ \sp{\ast} $-algebras and an application


Author: J. Vukman
Journal: Proc. Amer. Math. Soc. 91 (1984), 367-372
MSC: Primary 46K05
MathSciNet review: 744631
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Abstract: Let $ \mathcal{A}$ be a complex hermitian Banach $ * $-algebra with an identity element $ e$. Suppose there exists an additive function $ f:\mathcal{A} \to \mathcal{A}$ such that $ f\left( a \right) = - {a^ * }af\left( {{a^{ - 1}}} \right)$ holds for all normal invertible elements $ a \in \mathcal{A}$. We prove that in this case $ f$ is of the form $ f\left( a \right) = f\left( {ie} \right)k$, where $ a = h + ik$. Using this result we generalize S. Kurepa's extension of Jordan-Neumann characterization of pre-Hilbert space.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0744631-9
PII: S 0002-9939(1984)0744631-9
Keywords: $ * $-algebra, Banach $ * $-algebra, hermitian Banach $ * $-algebra, $ {B^ * }$-algebra, vector space, module, additive function, $ \mathcal{A}$-bilinear form, $ \mathcal{A}$-quadratic form
Article copyright: © Copyright 1984 American Mathematical Society