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: B loc a := x ∈ A : a n xa −n ≤ c x n α(x) for some α (x) ≥ 0 and c x > 0. In this article we study some properties of this algebra; in particular, we prove that B loc e+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 α, 0 ≤ α < 1, and consider the following subspace of A : D α a,b := {x ∈ A : a n xb n = O (n α) as n → ∞}. Here we study some properties of the subspaces D α a,b and D α b,a .