Continuity of spectral radius and type I $C^*$-algebras
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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.References
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Additional Information
- Tatiana Shulman
- Affiliation: Department of Mathematical Physics and Differential Geometry, Institute of Mathematics of Polish Academy of Sciences, 00-656 Warsaw, Poland
- MR Author ID: 684365
- Received by editor(s): August 8, 2017
- Received by editor(s) in revised form: February 23, 2018
- Published electronically: October 31, 2018
- Additional Notes: The research of the author was supported by a Polish National Science Centre grant under the contract number DEC- 2012/06/A/ST1/00256, by the grant H2020-MSCA-RISE-2015-691246-QUANTUM DYNAMICS, and by the Eric Nordgren Research Fellowship Fund at the University of New Hampshire.
- Communicated by: Adrian Ioana
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 641-646
- MSC (2010): Primary 46L05
- DOI: https://doi.org/10.1090/proc/14169
- MathSciNet review: 3894903