On Some Deddens Subspaces of Banach Algebras

Let A be a Banach algebra with a unit e, and let a is an element of A an invertible element. We define the following algebra: B-a(loc ):= {x is an element of A : parallel to a(n)xa(-n)parallel to <= c(x)n(alpha)((x) )for some alpha(x) >= 0 andc(x) 0}. In this article we study some properties of this algebra; in particular, we prove that B-e+p(loc) = {x is an element of A : Px (e - p) = 0}, where p is an idempotent in A. We also investigate the following Deddens subspace. Let a, b is an element of A be two elements. Fix any number alpha, 0 <= alpha < 1, and consider the following subspace of A: D-a,b(alpha ):= {x is an element of A: parallel to a(n)xb(n)parallel to = O(n(alpha)) as n -> infinity}. Here we study some properties of the subspaces D-a,b(alpha) and D-b,a(alpha).