Numerical Range in C*-Algebras

Abstract

Let A be a C*-algebra with unit 1 and let S be the state space of A, i.e., S = {ϕ ∈ A∗ : ϕ > 0, ϕ(1) = 1}. For each a ∈ A, the C*-algebra numerical range is defined by V (a) := {ϕ(a) : ϕ ∈ S}. We prove that if V (a) is a disc with center at the origin, then ka+a ∗ k = ka − a ∗ k