LOCAL DERIVATIONS OF NEST-ALGEBRAS

Let X be an arbitrary reflexive Banach space, and let N be a nest on X. Denote by D(N) the set of all derivations from AlgN into AlgN. For N subset of N, we set N_ = V{M is an element of N : M subset of N). We also write 0_ = 0. Finally, for E, F is an element of N define (E, F) = (K is an element of N : E subset of K subset of or equal to F}. In this paper we prove that a sufficient condition for D(N) to be (topologically) algebraically reflexive is that for all 0 not equal E is an element of N and for all X not equal F is an element of N, there exist M is an element of (0, E) and N is an element of (F, X), such that M_ subset of M and N_ subset of N. In particular, we prove that this condition automatically holds for nests acting on finite-dimensional Banach spaces.