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