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

 

The Arens product and duality in $ B\sp{\ast} $-algebras. II


Author: Pak-ken Wong
Journal: Proc. Amer. Math. Soc. 27 (1971), 535-538
MSC: Primary 46.60
MathSciNet review: 0275176
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Abstract: Let $ A$ be a commutative $ {B^ \ast }$-algebra, $ \Phi $ its carrier space and $ {A^ \ast }$ the conjugate space of $ A$. Let $ A'$ be the closed subspace of $ {A^ \ast }$ spanned by $ \Phi $. We show that $ A$ is a dual algebra if and only if $ A' = {A^ \ast }$ and for each $ x \in A$, the mapping $ {T_x}:f \to f \ast x$ is a weakly completely continuous operator on $ {A^ \ast }$. This improves an early result by B. J. Tomiuk and the author. A similar result holds for general $ {B^ \ast }$-algebras.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0275176-1
PII: S 0002-9939(1971)0275176-1
Keywords: Dual $ {B^ \ast }$-algebra, Arens product, Arens regular, carrier space, weakly completely continuous operators, selfadjoint minimal idempotents
Article copyright: © Copyright 1971 American Mathematical Society