Continuity of spectral radius and type I $C^*$-algebras

TL;DR

This paper characterizes type I $C^*$-algebras by the continuity of spectral radius and the properties of nilpotent elements, answering a question posed by Shulman and Turovskii.

Contribution

It establishes that spectral radius continuity and nilpotent element closure characterize type I $C^*$-algebras, providing a definitive answer to an open question.

Findings

Spectral radius is continuous iff the $C^*$-algebra is type I.

Closure of nilpotents contains an element with non-zero spectrum iff the algebra is not type I.

Answers a question of Shulman and Turovskii regarding spectral properties.

Abstract

It is shown that the spectral radius is continuous on a $C^*$-algebra if and only if the $C^*$-algebra is type I. This answers a question of V. Shulman and Yu.~Turovskii [10]. It is shown also that the closure of nilpotents in a $C^*$-algebra contains an element with non-zero spectrum if and only if the $C^*$-algebra is not type I.