Perturbation and Spectral Discontinuity in Banach Algebras

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 $z$.