Inner Derivations and Primal Ideals of C∗‐Algebras

Let A be a C∗‐algebra. For a ∈ A let D ( a, A ) denote the inner derivation induced by a , regarded as a bounded operator on A , and let d ( a, Z ( A )) denote the distance of a from Z ( A ), the centre of A . Let K ( A ) be the smallest number in [0, ∞] such that d ( a, Z ( A ))⩽ K ( A )∥ D ( a, A )∥ for all a ∈ A . It is shown that if A is non‐commutative and has an identity then either K ( A ) = 1 2 , or K ( A ) = 1 / √3, or K ( A ) ⩾ 1. Necessary and sufficient conditions for these three possibilities are given in terms of the primitive and primal ideals of A . If A is a quotient of an AW∗‐algebra then K ( A ) ⩽ 1 2 . Helly's Theorem is used to show that if A is a weakly central C∗‐algebra then K ( A ) ⩽ 1.