A Class Of Counterexamples Concerning an Open Problem

Kenneth R. Davidson raised ten open problems in the book Nest Algebras. One of these open problems is Problem 7 If $ {\cal K} \cap {\text{Alg}}{\cal L} Kenneth R. Davidson raised ten open problems in the book Nest Algebras. One of these open problems is Problem 7 If is weak. dense in , where is the set of all compact operators in , is completely distributive? In this note, we prove that there is a reflexive subspace lattice on some Hilbert space, which satisfies the following conditions: (a) is dense in in the ultrastrong operator topology, where is the set of all finite rank operators in ; (b) isn’t a completely distributive lattice. The subspace lattices that satisfy the above conditions form a large class of lattices. As a special case of the result, it easy to see that the answer to Problem 7 is negative.