On some deddens subspaces of Banach algebras

Abstract

Let A be a Banach algebra with a unit e, and let a ? A be an invertible element. We define the following algebra: Bloca := n {x ? A :??anxa-n?? ? cxn?(x) for some ?(x) ? 0 and cx > 0}. In this articlewestudy some properties of this algebra; in particular, weprove that Bloce+p = {x ? A : px (e-p) = 0}, where p is an idempotent in A. We also investigate the following Deddens subspace. Let a,b ? A be two elements. Fix any number ?,0 ? ? < 1, and consider the following subspace of A : D? a,b := {x ? A : ??anxbn?? = O(n?) as n ? ?}. Here we study some properties of the subspaces D?a,b and D?b,a.