Compact operators and the geometric structure of 𝐶*-algebras

Anoussis, M.; Katsoulis, E.

doi:10.1090/S0002-9939-96-03285-6

Compact operators and the geometric structure of $C^*$-algebras
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by M. Anoussis and E. G. Katsoulis

Proc. Amer. Math. Soc. 124 (1996), 2115-2122

DOI: https://doi.org/10.1090/S0002-9939-96-03285-6

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Abstract:

Given a $C^\ast$-algebra $\mathcal {A}$ and an element $A\in \mathcal {A}$, we give necessary and sufficient geometric conditions equivalent to the existence of a representation $(\phi ,\mathcal {H})$ of $\mathcal {A}$ so that $\phi (A)$ is a compact or a finite-rank operator. The implications of these criteria on the geometric structure of $C^\ast$-algebras are also discussed.

References

Kenneth R. Davidson, Nest algebras, Pitman Research Notes in Mathematics Series, vol. 191, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. Triangular forms for operator algebras on Hilbert space. MR 972978
R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–415. MR 203464, DOI 10.1090/S0002-9939-1966-0203464-1
J. A. Erdos, On certain elements of $C^{\ast }$-algebras, Illinois J. Math. 15 (1971), 682–693. MR 290120, DOI 10.1215/ijm/1256052521
P. Harmand, D. Werner, and W. Werner, $M$-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993. MR 1238713, DOI 10.1007/BFb0084355
Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Pure and Applied Mathematics, vol. 100, Academic Press, Inc., Orlando, FL, 1986. Advanced theory. MR 859186, DOI 10.1016/S0079-8169(08)60611-X
R. L. Moore and T. T. Trent, Extreme points of certain operator algebras, Indiana Univ. Math. J. 36 (1987), no. 3, 645–650. MR 905616, DOI 10.1512/iumj.1987.36.36036
James Rovnyak, Operator-valued analytic functions of constant norm, Czechoslovak Math. J. 39(114) (1989), no. 1, 165–168. MR 983493, DOI 10.21136/CMJ.1989.102288