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


Characterizations of $ w\sp{\ast} $-homomorphisms and expectations

Author: T. W. Palmer
Journal: Proc. Amer. Math. Soc. 46 (1974), 265-272
MSC: Primary 46K99
MathSciNet review: 0361804
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Abstract: If $ \mathfrak{A}$ and $ \mathfrak{B}$ are $ \ast $-algebras a map $ \phi :\mathfrak{A} \to \mathfrak{B}$ is called a Schwarz map if it is linear and satisfies the Cauchy-Schwarz inequality $ \phi {(a)^ \ast }\phi (a) \leq \phi (a ^\ast a)$ for all $ a \in \mathfrak{A}$. Under mild restrictions on $ \mathfrak{A}$ and $ \mathfrak{B}$, $ \ast $-homomorphisms and expectations are characterized in terms of Schwarz maps $ \phi :\mathfrak{A} \to \mathfrak{B}$. The proofs are based on an elementary result on the multiplicative properties of Schwarz maps.

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PII: S 0002-9939(1974)0361804-1
Keywords: $ \ast $-algebra, $ \ast $-homomorphism, Cauchy-Schwarz inequality, expectation, Banach $ \ast $-algebra, $ {U^ \ast }$-algebra, Jordan $ \ast $-homomorphism
Article copyright: © Copyright 1974 American Mathematical Society

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