Continuity of spectral radius and type I 𝐶*-algebras

Shulman, Tatiana

doi:10.1090/proc/14169

Continuity of spectral radius and type I $C^*$-algebras
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Proc. Amer. Math. Soc. 147 (2019), 641-646 Request permission

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. 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.

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