Perturbation and spectral discontinuity in Banach algebras
Volume 203 / 2011
Studia Mathematica 203 (2011), 253-263
MSC: Primary 46H05; Secondary 47A10, 47A55.
DOI: 10.4064/sm203-3-3
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 $x\in Y$ whose spectrum separates the plane there corresponds a perturbation of $x$, of the form $z=x+a$ where $a\in I$, such that the spectrum function on $A$ is discontinuous at $z$.