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
DOI:
https://doi.org/10.1090/S0002-9939-1970-0259620-0
MathSciNet review:
0259620
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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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Additional Information
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