# 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.

**DOI: **http://dx.doi.org/10.7900/jot.2012nov15.1983

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

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