Perturbation and spectral discontinuity in Banach algebras

Rudi Brits

Studia Mathematica (2011)

  • Volume: 203, Issue: 3, page 253-263
  • ISSN: 0039-3223

Abstract

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We extend an example of B. Aupetit, which illustrates spectral discontinuity for operators on an infinite-dimensional separable Hilbert space, to a general spectral discontinuity result in abstract Banach algebras. This can then be used to show that given any Banach algebra, Y, one may adjoin to Y a non-commutative inessential ideal, I, so that in the resulting algebra, A, the following holds: To each x ∈ Y whose spectrum separates the plane there corresponds a perturbation of x, of the form z = x + a where a ∈ I, such that the spectrum function on A is discontinuous at z.

How to cite

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Rudi Brits. "Perturbation and spectral discontinuity in Banach algebras." Studia Mathematica 203.3 (2011): 253-263. <http://eudml.org/doc/285494>.

@article{RudiBrits2011,
abstract = {We extend an example of B. Aupetit, which illustrates spectral discontinuity for operators on an infinite-dimensional separable Hilbert space, to a general spectral discontinuity result in abstract Banach algebras. This can then be used to show that given any Banach algebra, Y, one may adjoin to Y a non-commutative inessential ideal, I, so that in the resulting algebra, A, the following holds: To each x ∈ Y whose spectrum separates the plane there corresponds a perturbation of x, of the form z = x + a where a ∈ I, such that the spectrum function on A is discontinuous at z.},
author = {Rudi Brits},
journal = {Studia Mathematica},
keywords = {Banach algebra; spectrum; spectral continuity; spectral radius; perturbation; inessential ideals},
language = {eng},
number = {3},
pages = {253-263},
title = {Perturbation and spectral discontinuity in Banach algebras},
url = {http://eudml.org/doc/285494},
volume = {203},
year = {2011},
}

TY - JOUR
AU - Rudi Brits
TI - Perturbation and spectral discontinuity in Banach algebras
JO - Studia Mathematica
PY - 2011
VL - 203
IS - 3
SP - 253
EP - 263
AB - We extend an example of B. Aupetit, which illustrates spectral discontinuity for operators on an infinite-dimensional separable Hilbert space, to a general spectral discontinuity result in abstract Banach algebras. This can then be used to show that given any Banach algebra, Y, one may adjoin to Y a non-commutative inessential ideal, I, so that in the resulting algebra, A, the following holds: To each x ∈ Y whose spectrum separates the plane there corresponds a perturbation of x, of the form z = x + a where a ∈ I, such that the spectrum function on A is discontinuous at z.
LA - eng
KW - Banach algebra; spectrum; spectral continuity; spectral radius; perturbation; inessential ideals
UR - http://eudml.org/doc/285494
ER -

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