Local derivations of nest algebras

Abstract

Let X be an arbitrary reflexive Banach space, and let N \mathcal {N} be a nest on X. Denote by D ( N ) \mathcal {D}(\mathcal {N}) the set of all derivations from Alg ⁡ N \operatorname {Alg}\mathcal {N} into Alg ⁡ N \operatorname {Alg}\mathcal {N} . For N ⊂ N N \subset \mathcal {N} , we set N − = ∨ { M ∈ N : M ⊂ N } {N_ - } = \vee \{ M \in \mathcal {N}:M \subset N\} . We also write 0 − = 0 {0_ - } = 0 . Finally, for E , F ∈ N E, F \in \mathcal {N} define ( E , F ] = { K ∈ N : E ⊂ K ⊆ F } (E,F] = \{ K \in \mathcal {N}:E \subset K \subseteq F\} . In this paper we prove that a sufficient condition for D ( N ) \mathcal {D}(\mathcal {N}) to be (topologically) algebraically reflexive is that for all 0 ≠ E ∈ N 0 \ne E \in \mathcal {N} and for all X ≠ F ∈ N X \ne F \in \mathcal {N} , there exist M ∈ ( 0 , E ] M \in (0,E] and N ∈ ( F , X ] N \in (F,X] , such that M − ⊂ M {M_ - } \subset M and N − ⊂ N {N_ - } \subset N . In particular, we prove that this condition automatically holds for nests acting on finite-dimensional Banach spaces.