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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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