Czechoslovak Mathematical Journal, Vol. 62, No. 4, pp. 1101-1116, 2012

Finite spectra and quasinilpotent equivalence
in Banach algebras

Rudi M. Brits, Heinrich Raubenheimer

Rudi Brits, Heinrich Raubenheimer, Department of Mathematics, University of Johannesburg, Aucklandpark Campus, Aucklandpark 6000, South Africa, e-mail: rbrits@uj.ac.za, heinrichr@uj.ac.za

Abstract: This paper further investigates the implications of quasinilpotent equivalence for (pairs of) elements belonging to the socle of a semisimple Banach algebra. Specifically, not only does quasinilpotent equivalence of two socle elements imply spectral equality, but also the trace, determinant and spectral multiplicities of the elements must agree. It is hence shown that quasinilpotent equivalence is established by a weaker formula (than that of the spectral semidistance). More generally, in the second part, we show that two elements possessing finite spectra are quasinilpotent equivalent if and only if they share the same set of Riesz projections. This is then used to obtain further characterizations in a number of general, as well as more specific situations. Thirdly, we show that the ideas in the preceding sections turn out to be useful in the case of $C^*$-algebras, but now for elements with infinite spectra; we give two results which may indicate a direction for further research.

Keywords: finite rank elements, quasinilpotent equivalence, normal elements

Classification (MSC 2010): 46H05, 46H10


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