Derivations of CDC algebras
Let \(H\) be a Hilbert space and let \({\mathcal A}\) be a subalgebra of \(B(H)\). We say that a derivation \(\delta:{\mathcal A}\to B(H)\) is quasi-spatial if there exists a densely defined, closed linear operator \(T: \text{Dom}(T)\to H\) such that \(\text{Dom}(T)\) is invariant under every operator in \(\mathcal A\) and \(\delta(A)x = (TA -AT)x\) for all \(A\in \mathcal A\) and \(x\in \text{Dom}(T)\). The main result of the present paper characterizes quasi-spatial derivations on \(\mathcal A = \text{Alg}\,\mathcal L\), where \(\mathcal L\) is a completely distributive and commutative lattice on a separable Hilbert space \(H\). It states that a derivation \(\delta:\text{Alg}\,{\mathcal L}\to B(H)\) is quasi-spatial if and only if \(\delta(R)\) maps the kernel of \(R\) into the range of \(R\) for every \(R\) from the algebra generated by all rank one operators in \(\text{Alg}\,\mathcal L\). An analogous result for isomorphisms was obtained earlier by \textit{F.~Gilfeather} and \textit{R.~L.\ Moore} [J.~Funct.\ Anal.\ 67, 264--291 (1986; Zbl 0635.47040)].
scientific article; zbMATH DE number 5072938
scientific article