Chaotic Banach algebras

Abstract

We construct an infinite dimensional non-unital Banach algebra A and a ∈ A such that the sets {za n : z ∈ C, n ∈ N} and {(1 + a) n a : n ∈ N} are both dense in A, where 1 is the unity in the unitalization A # = A ⊕ span {1} of A. As a byproduct, we get a hypercyclic operator T on a Banach space such that T ⊕ T is non-cyclic and σ(T ) = {1}.