All-derivable points in nest algebras (Q972785)

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 \(\{0\}\neq M\neq H\) and write \(\widehat M\) for \(M\) or \(M^\perp\) . Our main result is: for any \(\Omega\in\text{alg}\,{\mathcal N}\) with \(\Omega=P(\widehat M)\Omega P(\widehat M)\), if \(\Omega|_{\widehat M}\) is invertible in \(\text{alg}\,{\mathcal N}_{\widehat M}\), then \(\Omega\) is an all-derivable point in \(\text{alg}\,{\mathcal N}\) for the strong operator topology.