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 Möbius transform $\varphi$ of the unit disk, then ${\cal A}$ is commutative.