We prove the following two improvements of a result of BECKER. (1) If A is a pro-C*-algebra, then every derivation on A is approximately inner. (2) If A is a separable a-C*-algebra, and ifevery C* quotient of A has the property that every derivation on it is inner, then also every derivation on A is inner. We also give an example of a derivation on a separable a-C*-algebra which is not inner but which induces an inner derivation on every C* quotient.
Derivations on pro-C*-algebras (inverse limits of C*-algebras; also called LMC*-algebras, generalized operator algebras, etc.) have been studied in [3]. It is shown there that if A is a pro-C*-algebra such that every derivation on each C* quotient of A is inner, then every derivation on A is approximately inner. We improve this result in two directions. First, we show that every derivation on A is approximately inner, with no conditions on the derivations on the C* quotients. Second, we show that if A is separable and metrizable, and if one does assume that every derivation on each C* quotient is inner, then in fact every derivation on A is inner. On the other hand we shown by example that it is possible to have a derivations on a separable o-C*-algebra which is not inner, but for which the induced derivation on the C* quotients are all inner. (Some of these C* quotients admit other derivations which are not inner.) ' ) Research partially supported by NSF grant DMS-91 06285. 16.