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Journal of the London Mathematical Society 1994 50(3):568-580; doi:10.1112/jlms/50.3.568
© 1994 by London Mathematical Society
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© The London Mathematical Society

Inner Derivations and Primal Ideals of C*-Algebras

Douglas W. B. Somerset

Department of Mathematics and Statistics, University of Newcastle-upon-Tyne Newcastle-upon-Tyne NE1 7RU

Received 27 October 1992.

Let A be a C*-algebra. For a isin 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, {infty}] such that d(a, Z(A))≤ K(A)||D(a, A)|| for all a isin A. It is shown that if A is non-commutative and has an identity then either K(A) = Formula, or K(A) = 1 / {surd}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) ≤ Formula. Helly's Theorem is used to show that if A is a weakly central C*-algebra then K(A) ≤ 1.


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