Local derivations of nest algebras

Abstract:

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