Numerical ranges of elements of involutive Banach algebras and commutativity

Abstract

Let $ {\cal A} $ be a unital involutive Banach algebra. Bonsall and Duncan defined the numerical range for each element x in $ {\cal A} $ by $ V(x) = \{f(x):f \in {\cal A'}, f(e) = 1 =|| f || \} $ , where e is the unit. We introduce another numerical range by $ W (x) = \{ f(x) : f \in {\cal A'}, f \geq 0, f(e) = 1 \} $ , and we call $ w(x) = {\rm sup} \{|z|: z \in W (x) \} $ the numerical radius of x. We give a few conditions for $ {\cal A} $ to be a C *-algebra, and we see that some mapping theorems for numerical ranges of elements of $ {\cal A} $ hold. We show that if w (x) ≤ 1 implies $ w (\varphi (x) ) \leq 1 $ for every Mobius transform $ \varphi $ of the unit disk, then $ {\cal A} $ is commutative.