2-Local ring derivations on nest algebras determined through rank-one operators

Abstract

UDC 517.98; 512.55

Local (2-local) derivations provide a flexible weakening of derivations, which have been extensively studied in $C^{*}$-algebras and nonselfadjoint operator algebras. For a nest algebra $\mathfrak{A} = \mathrm{Alg}\,\mathscr{N}$ acting on an infinite-dimensional Hilbert space, every local derivation is also a derivation provided that the Leibniz rule holds in the whole algebra. We develop a ring-theoretic refinement in which locality assumptions are imposed only on the rank-one operators. Let $\mathfrak{F}$ denote the ideal of finite-rank operators in $\mathfrak{A}$ and let $\mathfrak{R}$ be the set of rank-one operators in $\mathfrak{F}.$ An additive map $\Delta\colon \mathfrak{A}\to\mathfrak{A}$ is called rank-one 2-local (resp., rank-one local) if, for every pair $R_1,R_2\in\mathfrak{R}$ (resp., for every $R\in\mathfrak{R}$) there exists a ring derivation $D_{R_1,R_2}$ (resp., $D_R$) on $\mathfrak{A}$ such that $\Delta(R_j)=D_{R_1,R_2}(R_j)$ for $j=1,2$ (resp., $\Delta(R)=D_R(R)$). We prove that if $\Delta$ is an additive rank-one 2-local derivation that maps $\mathfrak{F}$ into itself, then $\Delta$ must be a complex linear derivation on the entire $\mathfrak{A},$ and, hence, inner. Moreover, every inner derivation of $\mathfrak{A}$ is rank-one local in this sense and  completely determined, up to central perturbations, by its values on $\mathfrak{R}.$ The argument relies on the structure of rank-one operators in nest algebras, the essentiality of $\mathfrak{F},$ and a careful comparison of the implementing derivations associated with different rank-one constraints. The stability of these conclusions under spatial and Lie ring isomorphisms between the nest algebras is also established.