Finite spectra and quasinilpotent equivalence in Banach algebras

Rudi M. Brits; Heinrich Raubenheimer

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 4, page 1101-1116
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Brits, Rudi M., and Raubenheimer, Heinrich. "Finite spectra and quasinilpotent equivalence in Banach algebras." Czechoslovak Mathematical Journal 62.4 (2012): 1101-1116. <http://eudml.org/doc/246954>.

@article{Brits2012,
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.},
author = {Brits, Rudi M., Raubenheimer, Heinrich},
journal = {Czechoslovak Mathematical Journal},
keywords = {finite rank elements; quasinilpotent equivalence; normal elements; quasinilpotent equivalence; finite rank element},
language = {eng},
number = {4},
pages = {1101-1116},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Finite spectra and quasinilpotent equivalence in Banach algebras},
url = {http://eudml.org/doc/246954},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Brits, Rudi M.
AU - Raubenheimer, Heinrich
TI - Finite spectra and quasinilpotent equivalence in Banach algebras
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 4
SP - 1101
EP - 1116
AB - 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.
LA - eng
KW - finite rank elements; quasinilpotent equivalence; normal elements; quasinilpotent equivalence; finite rank element
UR - http://eudml.org/doc/246954
ER -

References

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