文献类型：期刊文献

中文题名：一些Kadison-Singer代数的例子

英文题名：Some Examples of Kadison-Singer Algebras

作者：于嘉琪[1];李建奎[1]

机构：[1]华东理工大学理学院,上海200237

年份：2018

卷号：44

期号：6

起止页码：945

中文期刊名：华东理工大学学报（自然科学版）

外文期刊名：Journal of East China University of Science and Technology

收录：CSTPCD;;Scopus;北大核心:【北大核心2017】；CSCD:【CSCD_E2017_2018】；

语种：中文

中文关键词：CSL代数;Kadison-Singer代数;五角子空间格;半单代数

外文关键词：CSL algebra;Kadison-Singer algebra;pentagon subspace lattice;semi-simple algebra

摘要：设H是一个复的Hilbert空间。L={(0),L,M,K,H}是一个五角子空间格,满足M是L和x0的闭线性扩张,其中非零元x0属于K┴但不属于K+L。本文证明了Alg L是一个半单的Kadison-Singer代数。同时还给出两个不是套代数的CSL代数的例子,一个相似于某个Kadison-Singer代数,另一个却与任何一个Kadison-Singer代数都不相似。
Kadison-Singer algebras are a new class of non-self-adjoint reflexive operator algebras on a Hilbert space,which make connections between theories of self-adjoint algebras and non-self-adjoint algebras.Kadison-Singer algebras are closely associated with von Neumann algebras,and they are full of topological structures.Such class of operator algebras provide a new way to develop the theories of self-adjoint algebras.Pentagon subspace lattices and double triangle lattices are playing important roles in theories of lattices,and most of the focus was only put on double triangle lattices before.So we begin the study in pentagon subspace lattices to show more examples of Kadison-Singer algebras.Since any pentagon subspace lattice with gap-dimension1is reflexive,we construct a pentagon subspace lattice with gap-dimension1to ensure the reflexivity of the algebra.Let H be a complex Hilbert space.Suppose L={(0),L,M,K,H}is a pentagon subspace lattice satisfying that M is the closed linear span of L and x0,where x0is a nonzero vector in K^┴but not in K+L.We prove Alg L is a semi-simple Kadison-Singer algebra.Besides,we know that nest algebras are Kadison-Singer algebras and nest algebras are invariant under similar transformation.However,any CSL algebra,not nest algebra,is not a Kadison-Singer algebra,and an algebra,which is similar to a Kadison-Singer,is not necessarily a Kadison-Singer algebra.So there are two natural questions:whether there exist a CSL algebra,not nest algebra,which can be similar to a Kadison-Singer algebra,and whether a CSL algebra exist but is not similar to any Kadison-Singer algebra.