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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The Arens product and duality in $B^{\ast }$-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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Keywords: Dual <!– MATH ${B^{\ast }}$ –> <IMG WIDTH="31" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${B^{\ast }}$">-algebra, Arens product, <!– MATH ${B^{\ast }}(\infty )$ –> <IMG WIDTH="67" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="${B^{\ast }}(\infty )$">-sum, multiplier algebra, strict topology, carrier space, compact operators
Article copyright: © Copyright 1970 American Mathematical Society