Continuity of spectral radius and type I $C^{\ast }$-algebras

Abstract

It is shown that the spectral radius is continuous on a $C^{\ast }$-algebra if and only if the $C^{\ast }$-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 $C^{\ast }$-algebra contains an element with non-zero spectrum if and only if the $C^{\ast }$-algebra is not type I.

arXiv commentThanks to grant H2020-MSCA-RISE-2015-691246-Quantum Dynamics and Polish Government grant 3542/H2020/2016/2 are added