Uniqueness and spectral variation in Banach algebras

Abstract

Let A be a complex semisimple Banach algebra with identity. We explore the situation whereby a portion of the elements of A have the same spectrum under multiplication by α ∈ A, as under multiplication by b ∈ A; and when this situation implies that α and b are the same. In particular we show that if the spectrum of ax equals the spectrum of bx for all x with a spectral radius away from the identity less than 1, then a and b coincide. By way of examples we show that this is the best situation possible in general. In another result we show that in the case where these spectra are nite, the assumption need only hold for an arbitrarily small open set in A, with the same conclusion. Additive versions of these results are also discussed.

Keywords: Banach algebra, spectrum, spectral radius, spectral variation, subharmonic, socle

Quaestiones Mathematicae 36(2013), 155-165