Derivations of subhomogeneous $C^*$-algebras are implemented by local multipliers

Abstract

A condition on a derivation of an arbitrary C*-algebra is presented entailing that it is implemented as an inner derivation by a local multiplier. It is an outstanding open question whether every derivation of a C*-algebra A can be implemented as an inner derivation by a local multiplier, that is, an element in the direct limit of the multiplier algebras of the closed essential ideals of A. An affirmative answer was given by Elliott [4] for AF-algebras, and by Pedersen [11] for general separable C*-algebras. In fact, it suffices to assume that every closed essential ideal of A is σ-unital; hence Pedersen's result entails Sakai's theorem that every derivation of a simple unital C*-algebra is inner. But only an affirmative answer in the non-separable case would cover, extend and unify the results that every derivation of a simple C*-algebra is inner in the multiplier algebra [13] and that all derivations of von Neumann algebras [6], [12] and AW*-algebras [10] are inner. This quest becomes even more attractive by the recent results in [9] and [14] implying that, if a derivation δ on A is inner in the multiplier algebra, then there is a local multiplier a of A implementing δ such that δ = 2 a. No progress on the above question seems to have been made since it was raised in [11] (see also [4]). The purpose of this note is to present a criterion on a given derivation δ of a (possibly non-separable) C*-algebra A implying that δ is inner in the local multiplier algebra M loc (A). Though this criterion, inspired by Herstein's work [5], is rather algebraic in nature, it is hoped that some approximate version may eventually yield a positive solution of the general problem. 1. Notation and preliminaries Throughout this paper, M (A) will denote the multiplier algebra of the C*-algebra A. A left ideal L of A is said to be essential if its left annihilator L ⊥ = {a ∈ A | aL = 0} is zero. For a (closed) two-sided ideal I, the left annihilator coincides with the right annihilator, and I + I ⊥ is a (closed) essential ideal. Given two closed essential ideals I, J in A such that J ⊆ I, J is an essential ideal in M (I) and hence M (I) embeds isometrically into M (J). Forming the C*-direct limit of the directed family of multiplier algebras so obtained yields the local multiplier algebra of A,

Inner Derivations and Weak-2-Local Derivations on the $$\hbox {C}^*$$ C ∗ -Algebra $$\varvec{C_0(L,A)}$$

“…G.K. Pedersen proved that for a separable C * -algebra A every derivation on A extends to an inner derivation of M loc (A) (see [27]). There are some recent extensions of these results due to D.W. Somerset [35] and I. Gogić [16,17]. But much less seems to be known about proper extensions of Sakai, i.e.…”