Local Derivations of Nest Algebras

Abstract

Let X be an arbitrary reflexive Banach space, and let X be a nest on X. Denote by 9'(X) the set of all derivations from AlgV into AlgAV. For N C X, we set N_ = V{M E X: M C N}. We also write 0= 0. Finally, for E, F eX define (E, F] = {K EX: E C K C F}. In this paper we prove that a sufficient condition for 2 (X) to be (topologically) algebraically reflexive is that for all 0 :& E E X and for all X # F E X , there exist M E (0, E] and N E (F, X], such that M_ C M and N_ C N. In particular, we prove that this condition automatically holds for nests acting on finite-dimensional Banach spaces.