Approximants, Commutants and Double Commutants in Normed Algebras
Approximants, Commutants and Double Commutants in Normed Algebras
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DOI：
10.1112/jlms/s2-25.3.499
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发表时间：
1982-06
影响因子：
1.2
通讯作者：
M. S. Lambrou
中科院分区：
文献类型：
--
作者：
M. S. Lambrou
本文的目的是证明一些关于算子代数的结果，这些算子代数使赋范空间的子空间的强自反格的元素保持不变（定义如下）。这些算子代数包括子空间的完备原子布尔格不变的代数类（Halmos在[5]和[6]中引入）和套代数（Ringrose在[17]中引入），因此包括$8 {X\X上有界算子代数，以及Kadison和Singer [8]的超可约极大三角代数。本文有五个主要结果。事实证明，我们考虑的代数类有丰富的秩一算子。事实上，第一个主要结果，定理3.1，表明，每个运营商在代数可以近似在一定意义上的有限和秩1运营商的代数。第二个主要结果，定理4.2，表明对于完备原子布尔子空间格，其不变算子代数的每个有限秩算子都可以写成代数中秩1算子的有限和（对于希尔伯特空间的情况，参见Longstaff [13]）。定理5.1和5.2首先给出了强自反格的不变算子代数的交换子的特征，其次给出了强自反格的不变算子代数的双交换子的特征。最后，第五个主要结果，定理5.8，给出了其不变算子代数是交换的基本格的充分必要条件（见[11]）。注意，在[9]中给出了强自反格中完备原子布尔子空间格的刻画，在[10]中给出了抽象刻画。
The purpose of this paper is to prove some results about the algebra of operators which leave invariant the elements of a strongly reflexive lattice of subspaces of a normed space (definitions are given below). These algebras of operators include the class of algebras which leave invariant a complete atomic Boolean lattice of subspaces (introduced by Halmos in [5] and [6]) and nest algebras (introduced by Ringrose in [17]) and so include $8 {X\the algebra of bounded operators on X, and the hyper-reducible maximal triangular algebras of Kadison and Singer [8]. The paper has five main results. It turns out that the class of algebras we consider has a rich supply of rank one operators. In fact the first main result, Theorem 3.1, shows that every operator in the algebra can be approximated in a certain sense by a finite sum of rank one operators from the algebra. The second main result, Theorem 4.2, shows that for complete atomic Boolean subspace lattices every finite rank operator of its algebra of invariant operators can be written as a finite sum of rank one operators from the algebra (for the Hilbert space case see Longstaff [13]). Theorems 5.1 and 5.2 give characterisations firstly of the commutant and secondly of the double commutant of the algebra of invariant operators of a strongly reflexive lattice. Finally the fifth main result, Theorem 5.8, gives necessary and sufficient conditions on the underlying lattice for its algebra of invariant operators to be abelian (see [11]). Note that a characterisation of complete atomic Boolean subspace lattices among strongly reflexive lattices appears in [9] and an abstract characterisation appears in [10].