An example on ordered Banach algebras

Abstract

Let B {\mathcal B} be a complex unital Banach algebra. We consider the Banach algebra A = B × C {\mathcal A}={\mathcal B} \times \mathbb C ordered by the algebra cone K = { ( a , ξ ) ∈ A : ‖ a ‖ ≤ ξ } K=\{(a,\xi ) \in {\mathcal A}: \|a\| \le \xi \} , and investigate the connection between results on ordered Banach algebras and the right bound of the numerical range of elements in B {\mathcal B} .