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Weakly completely continuous elements of $ C\sp{\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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0383095-9
Keywords: Dual $ {C^{\ast}}$-algebra, weakly completely continuous element, compact element, compact operator
Article copyright: © Copyright 1975 American Mathematical Society

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