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Journal of Operator Theory

Volume 72, Issue 1, Summer 2014  pp. 135-158.

Normal limits of nilpotent operators in $C^*$-algebras

Authors:  Paul Skoufranis
Author institution: Department of Mathematics, UCLA, Los Angeles, California, 90095-1555 U.S.A.

Summary: We will investigate the intersection of the normal operators with the closure of the nilpotent operators in various $C^*$-algebras. A complete description of the intersection will be given for unital, simple, purely infinite $C^*$-algebras. The intersection in AF $C^*$-algebras is also of interest. In addition, an example of a separable, nuclear, quasidiagonal $C^*$-algebra where every operator is a limit of nilpotent operators will be constructed.

Keywords: $C^*$-algebra, nilpotent operators, quasinilpotent operators, normal operator, norm-limit, purely infinite $C^*$-algebra

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