A class of counterexamples concerning an open problem (Q2577858)

Let \(H\) be a Hilbert space and \(B(H)\) be the algebra of all bounded linear operators on \(H\). For any lattice of subspaces \(L\), write the set \[ \text{Alg}{L}=\{A \in B(H) : AY \subseteq Y \, \forall Y \in L \}. \] The main result of this paper is the following Theorem. There exists a reflexive subspace lattice \(L\) on some Hilbert space which satisfies the following conditions: (1) \(\mathcal{F}(\text{Alg}L)\) is dense in \(\text{Alg}{L}\) in the ultrastrong operator topology, where \(\mathcal{F}(\text{Alg}{L})\) is the set of all finite rank operators in \(\text{Alg}{L}\); (2) \(L\) is not completely distributive.