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Finite spectra and quasinilpotent equivalence in Banach algebras. (English) Zbl 1274.46094
The notion of quasinilpotent equivalence was introduced by I. Colojoară and C. Foiaş [Theory of generalized spectral operators. Mathematics and its Applications. 9. New York-London-Paris: Gordon and Breach Science Publishers (1968; Zbl 0189.44201)].
It is well known that the quasinilpotent equivalence preserves the spectrum. The present paper shows that quasinilpotent equivalent elements with finite spectrum in a semisimple Banach algebra even have equal Riesz projections. In particular, quasinilpotent equivalent elements in the socle have the same spectral multiplicities, traces and determinants.
The paper further studies the quasinilpotent equivalence for normal elements in \(C^{*}\)-algebras with finite or infinite spectra.
Reviewer: V. Müller (Praha)

46H05 General theory of topological algebras
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