Derivations of quasitriangular algebras (Q1072092)

A nest is a family of projections in L(H) which is linearly ordered, contains 0 and I, and is closed in the strong operator topology. Given a nest \({\mathcal P}\), let alg \({\mathcal P}=\{T\in L(H):\) \(P^{\perp}TP=0\), \(\forall P\in {\mathcal P}\}\). alg \({\mathcal P}\) is the nest algebra associated with \({\mathcal P}\). Let QT(\({\mathcal P})=alg {\mathcal P}+{\mathcal K}\), where \({\mathcal K}\) is the set of all compact operators. QT(\({\mathcal P})\) is the quasitriangular algebra associated with \({\mathcal P}.\) The main result is the following: Any automorphism \(\alpha\) of a quasitriangular algebra with \(\| \alpha -id\| <1\) must be inner. It follows that every derivation on a quasitriangular algebra is inner.