Derivations Determined by Multipliers on Ideals of a C*-Algebra

Abstract

Sakai's theorem that every derivation of a simple C*-algebra is determined by a multiplier is generalized, in the class of separable approximately finite-dimensional C*-algebras, as follows. It is shown that, in such a C*-algebra, any derivation can be approximated arbitrarily closely in norm by a derivation which is determined by a multiplier on a nonzero closed two-sided ideal. It is shown, moreover, that the multiplier may be chosen to have norm bounded by fixed multiple of the norm of the derivation.