A Class Of Counterexamples Concerning an Open Problem
PeiXinCHEN, ShiJieLU · Acta Scientiarum Naturalium Universitatis Sunyatseni · 2005
Kenneth R. Davidson raised ten open problems in the book Nest Algebras. One of theseopen problems isProblem 7 If K（交集）AlgL is weak^* dense in AlgL, where K is 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 AlgL in the ultrastrong operator topology, where F(AlgL) is the set of all finite rank operators in AlgL; (b) 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.