Centralizers for subsets of normed algebras

Abstract

Let G be the set of invertible elements of a normed algebra A with an identity. For some but not all subsets H of G we have the following dichotomy. For x ∈ A either $cxc^{-1} = x$ for all c ∈ H or $sup {∥cxc^{-1}∥ : c ∈ H} = ∞$. In that case the set of x ∈ A for which the sup is finite is the centralizer of H