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An example is given to show that, in a C -algebra with- out unit, a derivation is not necessarily determined by a multiplier in some quotient algebra. When one deals with the derivations of a C*-algebra without unit, a reasonable replacement for an inner derivation is the derivation determined by a multiplier. By the multiplier algebra MIA) of a C*-algebra A, we mean the idealizer of A in the universal enveloping von Neumann algebra A (i.e. the second dual of A). It happens, however, that the definition does not depend on the von Neumann algebra which contains A; that is, the idealizer in that von Neumann algebra is isomorphic to MIA) by an isomor- phism which fixes all elements of A (l, Proposition 2.4). Let 5 be a deri- vation of A. We say that ?5 is determined by a multiplier if it becomes in- ner when it is extended to the derivation of MIA), i.e. if we can choose a multiplier of A as a generator of ?5. In general, a derivation may not be de- termined by a multiplier. However, one may ask whether the restriction of ?5 to some small ideal is determined by a multiplier. An example for which the an- swer to this question is negative is shown in (2, 6.5). Now there arises another basic question for <5: whether we can find an ideal / such that the induced derivation of ?5 in the quotient algebra A/I is determined by a multiplier. When the algebra has a unit this is true by Sakai's theorem on derivations of simple C*-algebras (4). However, we show in the following that the an- swer to this question is negative in general. The author is indebted to C. A. Akemann for pointing out the following example of Dixmier-Behncke- Krauss-Leptin (3).

Derivations of 𝐶*-algebras which are not determined by multipliers in any quotient algebra