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 KAlgLis weak. dense in AlgL, where Kis the set of all compact operators in B(H), is L completely distributive? In this note, we prove that there is a reflexive subspace lattice L on some Hilbert space, which satisfies the following conditions: (a) F(AlgL)is dense in AlgLin the ultrastrong operator topology, where F(AlgL)is the set of all finite rank operators in AlgL; (b) Lisnt 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.