A note on the compact elements of $C^{\ast }$-algebras
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- by Kari Ylinen
- Proc. Amer. Math. Soc. 35 (1972), 305-306
- DOI: https://doi.org/10.1090/S0002-9939-1972-0296716-3
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Abstract:
It is shown that for any ${C^ \ast }$-algebra A there is a faithful representation $\pi$ of A on a Hilbert space H such that, for $u \in A$, the map $x \mapsto uxu$ is a compact operator on A if and only if $\pi (u)$ is a compact operator on H.References
- J. C. Alexander, Compact Banach algebras, Proc. London Math. Soc. (3) 18 (1968), 1–18. MR 229040, DOI 10.1112/plms/s3-18.1.1
- J. A. Erdos, On certain elements of $C^{\ast }$-algebras, Illinois J. Math. 15 (1971), 682–693. MR 290120, DOI 10.1215/ijm/1256052521
- Charles E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115101
- Klaus Vala, On compact sets of compact operators, Ann. Acad. Sci. Fenn. Ser. A I 351 (1964), 9. MR 0169078
- Kari Ylinen, Compact and finite-dimensional elements of normed algebras, Ann. Acad. Sci. Fenn. Ser. A I 428 (1968), 37. MR 0238089
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 35 (1972), 305-306
- MSC: Primary 46L05
- DOI: https://doi.org/10.1090/S0002-9939-1972-0296716-3
- MathSciNet review: 0296716