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.
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