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_{a}^{loc}:={x∈A:‖aⁿxa⁻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_{e+p}^{loc}={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_{a,b}^{α}:={x∈A:‖aⁿxbⁿ‖=O(n^{α}) as n→∞}.

Here we study some properties of the subspaces D_{a,b}^{α} and D_{b,a}^{α}.