Let A be a Banach algebra with a unit e, and let a ∈ A be an invertible element. We define the following algebra: Bloca := { x ∈ A : ∥∥∥anxa−n∥∥∥ ≤ cxnα(x) for some α (x) ≥ 0 and cx > 0} . In this article we study some properties of this algebra; in particular, we prove thatBloce+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 ofA : Dαa,b := {x ∈ A : ‖anxbn‖ = O (nα) as n→∞} . Here we study some properties of the subspaces Dαa,b and D α b,a.