A class of counterexamples concerning an open problem

被引：7

作者：

Chen, PX ^{[1
]}

Lu, SM

机构：

[1] Nanjing Univ Sci & Technol, Dept Math, Nanjing 210094, Peoples R China

[2] Zhejiang Univ, Dept Math, Hangzhou 310027, Peoples R China

来源：

ACTA MATHEMATICA SINICA-ENGLISH SERIES | 2005年 / 21卷 / 01期

关键词：

completely distributive subspace lattice; ultrastrong topology; counterexample;

D O I：

10.1007/s10114-004-0397-0

中图分类号：

O29 [应用数学];

学科分类号：

070104 [应用数学];

摘要：

Kenneth R. Davidson raised ten open problems in the book Nest Algebras. One of these open problems is Problern 7 If K boolean AND 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 T(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.

引用

页码：9 / 12

页数：4