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Proceedings of the American Mathematical Society

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

 
 

 

Weakly completely continuous elements of $C^{\ast }$-algebras


Author: Kari Ylinen
Journal: Proc. Amer. Math. Soc. 52 (1975), 323-326
MSC: Primary 46L05; Secondary 47B05
DOI: https://doi.org/10.1090/S0002-9939-1975-0383095-9
MathSciNet review: 0383095
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Abstract: For a $C^{\ast }$-algebra $A$ and $u\epsilon A$, the equivalence of the following three statements is proved: (i) the map $x \mapsto uxu$ is a compact operator on $A$, (ii) (resp. (iii)) the map $x \mapsto ux$ (resp. $x \mapsto xu$) is a weakly compact operator on $A$. The canonical image of a dual ${C^{\ast }}$-algebra $A$ in its bidual ${A^{{\ast }{\ast }}}$ is characterized as the set of the weakly completely continuous elements of ${A^{{\ast }{\ast }}}$.


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Keywords: Dual <!– MATH ${C^{\ast }}$ –> <IMG WIDTH="31" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img11.gif" ALT="${C^{\ast }}$">-algebra, weakly completely continuous element, compact element, compact operator
Article copyright: © Copyright 1975 American Mathematical Society