A Class Of Counterexamples Concerning an Open Problem

Abstract

Kenneth R. Davidson raised ten open problems in the book Nest Algebras. One of these open problems is

Problem 7 If ${\cal K} \cap {\text{Alg}}{\cal L}$is weak. dense in ${\text{Alg}}{\cal L}$, where ${\cal K}$is the set of all compact operators in ${\cal B}({\cal H})$, is ${\cal L}$ completely distributive?

In this note, we prove that there is a reflexive subspace lattice ${\cal L}$ on some Hilbert space, which satisfies the following conditions:

(a) ${\cal F}({\text{Alg}}{\cal L})$is dense in ${\text{Alg}}{\cal L}$in the ultrastrong operator topology, where ${\cal F}({\text{Alg}}{\cal L})$is the set of all finite rank operators in ${\text{Alg}}{\cal L}$;

(b) ${\cal L}$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.