Inner Derivations and Primal Ideals of C*-Algebras

Let A be a C*-algebra. For a A let D(a, A) denote the innerderivation 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 centreof 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 isnon-commutative and has an identity then either K(A) = , or K(A) = 1 / 3, or K(A) 1. Necessaryand sufficient conditions for these three possibilities aregiven in terms of the primitive and primal ideals of A. If Ais a quotient of an AW*-algebra then K(A) . Helly's Theorem is used to show that if A is aweakly central C*-algebra then K(A) 1.