All-derivable points in nest algebras

Abstract

Suppose that $\mathcal{A}$ is an operator algebra on a Hilbert space $H$. An element $V$ in $\mathcal{A}$ is called an all-derivable point of $\mathcal{A}$ for the strong operator topology if every strong operator topology continuous derivable mapping $\varphi$ at $V$ is a derivation. Let $\mathcal{N}$ be a complete nest on a complex and separable Hilbert space $H$. Suppose that $M$ belongs to $\mathcal{N}$ with $\{0\}\ne M\ne H$ and write $\hat{M}$ for $M$ or $M^{\perp }$. Our main result is: for any $\Omega \in alg\mathcal{N}$ with $\Omega =P(\hat{M})\Omega P(\hat{M})$, if $\Omega {|}_{\hat{M}}$ is invertible in $alg{\mathcal{N}}_{\hat{M}}$, then $\Omega$ is an all-derivable point in $alg\mathcal{N}$ for the strong operator topology.

12 pages, latex