On topological reflexivity of the spaces of derivations on operator algebras (Q1607515)

For a fixed complex Banach space \(X\), let \(L(X)\) and \(B(X)\) denote the algebras of all linear and bounded linear operators on \(X\), respectively. For a lattice \({\mathcal L}\) of subspaces of \(X\) and \(N\in {\mathcal L}\), let \(N_{-}=\bigvee\{M\in {\mathcal L} :N\not\subseteq M\}\) and \(N_{+}=\bigwedge\{ M\in {\mathcal L}:M\not\subseteq N\}\). The authors prove that: (i) If \({\mathcal M}\) is a von Neumann algebra in a Hilbert space \(H\), then the space of bounded derivations on \({\mathcal M}\) is topologically reflexive in \(B({\mathcal M})\) in the weak operator topology. (ii) If \({\mathcal B}\) is a reflexive algebra in the Banach space \(X\) such that both \(0_{+}\neq 0\) and \(X_{-}\neq X\) are in \(\operatorname {Lat}{\mathcal B}\) , then the space of derivations on \({\mathcal B}\) is topologically reflexive in \(L({\mathcal B})\) in the weak operator topology.