Finite Spectra and Quasinilpotent Equivalence in Banach Algebras

TL;DR

This paper explores quasinilpotent equivalence in Banach algebras, showing it implies spectral equality and other invariants for socle elements, and characterizes equivalence via Riesz projections for elements with finite spectra.

Contribution

It introduces new characterizations of quasinilpotent equivalence, including spectral and Riesz projection conditions, extending to elements with infinite spectra in $C^*$-algebras.

Findings

Quasinilpotent equivalence implies spectral equality and matching trace, determinant, and multiplicities.

Equivalence for finite spectrum elements is characterized by shared Riesz projections.

Results suggest potential for further research in $C^*$-algebras with infinite spectra.

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.