A Class Of Counterexamples Concerning an Open Problem  

作　　者：PeiXinCHEN ShiJieLU 

机构地区：[1]DepartmentofMathematics,NanjingUniversityofScienceandTechnology,Nanjing210094,P.R.China [2]DepartmentofMathematics,ZhejiangUniversity,Hangzhou310027,P.R.China

出　　处：《Acta Mathematica Sinica,English Series》2005年第1期9-12,共4页数学学报（英文版）

摘　　要：If K ∩ AlgL is weak. dense in AlgL, where K is the set of all compactoperators in B(H), is completely distributive? In this note, we prove that there is a reflexivesubspace lattice L on some Hilbert space, which satisfies the following conditions: (a) F(AlgL) isdense in AlgL in the ultrastrong operator topology, where F(AlgL) is the set of all finite rankoperators in AlgL; (b) L isnt a completely distributive lattice. The subspace lattices that satisfythe above conditions form a large class of lattices. As a special case of the result, it easy to seethat the answer to Problem 7 is negative.If K ∩ AlgL is weak. dense in AlgL, where K is the set of all compactoperators in B(H), is completely distributive? In this note, we prove that there is a reflexivesubspace lattice L on some Hilbert space, which satisfies the following conditions: (a) F(AlgL) isdense in AlgL in the ultrastrong operator topology, where F(AlgL) is the set of all finite rankoperators in AlgL; (b) L isnt a completely distributive lattice. The subspace lattices that satisfythe above conditions form a large class of lattices. As a special case of the result, it easy to seethat the answer to Problem 7 is negative.

关 键 词：Completely distributive subspace lattice Ultrastrong topology COUNTEREXAMPLE 

分 类 号：O177.1[理学—数学]