Derived $C^{\ast}$-algebras of primitive $C^{\ast}$-algebras

Abstract

Introduction.It has been known that every derivation of a T7*-algebra is inner (as a corollary, every derivation of a C*-algebra is weakly inner), and every derivation of a simple C*-algebra with identity is inner (cf.[6]).Moreover it has been shown that for a simple C*algebra SI with or without identity, there exists a unique primitive C*algebra ®(2I) with identity (called the derived C*-algebra of SI) such that (1) SI is an ideal of 2)(SI); (2) for every derivation δ of SI, there is a unique (modulo scalar multiples of identity) elementThese results make the study of derivations in general C*-algebras, more or less, possible to reduce to the study of derivations in simple C*-algebras if the C*-algebras have only maximal ideals as primitive ideals.However there are many C*-algebras which do not have any maximal ideal ([3]).For the study of derivations in these C*-algebras, it is desirable to analyse derivations in primitive C*-algebras.In the present paper, we shall generalize the notion of derived C*algebras to primitive C*-algebras to make possible to reduce the study of derivations in general C*-algebras to the study of derivations in primitive C*-algebas.We shall explain briefly the main result in this paper.Let SI be a primitive C*-algebras (more generally, a factorial C*-algebra) and let JD(SI) be the Lie algebra of all derivations on SI.For an arbitrary faithful factorial ^representation {TΓ, £} of SI on a Hubert space X, it is known that a unique (modulo scalar multiples of identity) element d δ in the weak closure τr(SI) of ττ(SI) such that π(<5(α)) = [d δ ,π(a)\ (αeSI).Now we shall identify SI with π(SI), and let ®^(SI) be the C*-subalgebra of B(X) generated by {d δ I d e D(3I)} and l x .Then it is easily imagined that the C*-algebra S) π (SI) is closely related to the structure of the Lie algebra Z?(SI) and so we may apply the C*-algebra theory to the study of D(3I).However