Numerical ranges of elements of involutive Banach algebras and commutativity

摘要

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