CSL代数加子群中的有限秩算子

Finite rank operators in additive subgroup of CSL algebras.

设L是Hilbert空间H上的一个交换子空间格(简记为CSL),引进了性质P并得到两个主要结果:(a)若G是一个具有性质P的加群,F∈G是一个可写作AlgL中有限个秩一算子之和的有限秩算子,那么,它一定可写作Ringrose理想R(L)中有限个秩一算子的和.(b)设M AlgL是一个具有性质P的左(右)(L)″-模,则M中的所有有限秩算子都包含在R1(L)‖·‖1,其中R1(L)代表由Ringrose理想中所有秩一算子生成的代数,‖·‖1是迹范数.

Let L be a CSL on H, introduce the concept of property P and obtain two main results: (a) Suppose G is an additive subgroup having the property P and F ∈G is a finite rank operator which can be written as a finite sum of rank one operators in AlgL. Then it can be written as a finite sum of rank one operators in the Ringrose ideal R (L). (b) Suppose M Alg L is a left (right) (L″)- module with property P. Then all finite rank operators in M are contained in  R 1 (L) ‖·‖1 ,where R1 (L) is the algebra generated by all rank one operators in the Ringrose ideal R (L), and ‖·‖1 is the trace norm.