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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


Authors: B. J. Tomiuk and Pak-ken Wong
Journal: Proc. Amer. Math. Soc. 25 (1970), 529-535
MSC: Primary 46.60
MathSciNet review: 0259620
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Abstract: Let $ A$ be a $ {B^{\ast}}$-algebra, $ {A^{{\ast}{\ast}}}$ its second conjugate space and $ \pi $ the canonical embedding of $ A$ into $ {A^{{\ast}{\ast}}}$. $ {A^{{\ast}{\ast}}}$ is a $ {B^{\ast}}$-algebra under the Arens product. Our main result states that $ A$ is a dual algebra if and only if $ \pi (A)$ is a two-sided ideal of $ {A^{{\ast}{\ast}}}$. Gulick has shown that for a commutative $ A,\;\pi (A)$ is an ideal if and only if the carrier space of $ A$ is discrete. As this is equivalent to $ A$ being a dual algebra, Gulick's result thus carries over to the general $ {B^{\ast}}$-algebra.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1970-0259620-0
PII: S 0002-9939(1970)0259620-0
Keywords: Dual $ {B^{\ast}}$-algebra, Arens product, $ {B^{\ast}}(\infty )$-sum, multiplier algebra, strict topology, carrier space, compact operators
Article copyright: © Copyright 1970 American Mathematical Society