Rank One Subspaces of Bimodules over Maximal Abelian Selfadjoint Algebras
Rank One Subspaces of Bimodules over Maximal Abelian Selfadjoint Algebras
复制标题
最大阿贝尔自伴代数双模的一阶子空间
DOI：
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发表时间：
1998
期刊：
影响因子：
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通讯作者：
V. Shulman
中科院分区：
文献类型：
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作者：
J. Erdos;A. Katavolos;V. Shulman
极大交换自伴代数上的左、右模算子空间（简称Masa双模）是交换子空间格代数的自然推广。本文研究了有限秩算子和有限秩模中各类紧算子的密度性质。证明了模闭masa双模M的有限秩算子都在M的秩1子空间的迹模闭包中。一个重要的结果是强自反masa双模（即，一个是它的秩1算子的自反船体）的秩1子空间在弱算子拓扑的模中是稠密的。然而，在对比的情况下，代数，它表明，这样的密度不需要举行的超弱拓扑。介绍了一种新的表示masa双模的方法。这使用了ω-拓扑的新概念。利用ω-支撑的概念，建立了自反masa双模与其ω-支撑之间的对应关系.它表明，如果一个C2-闭masa双模包含一个迹类算子，那么它必须包含秩1算子;事实上，每个这样的操作是在C2-范数闭包秩1子空间的模块。因此，迹类算子的任何masa双模的弱闭包都是强自反的。然而，秩一子空间的迹范数闭包不需要包含模的所有迹类算子。还证明了存在不含迹类算子但含有属于Cp的算子的CSL代数，其中allp>1.由此可以得出，一个由秩为1的算子所张成的传递双模在Cp中对于1 p<∞不一定是稠密的。作为应用，证明了存在一个交换子空间格L，使得L是非合成的，但每个包含masa且具有不变格L的弱闭代数都与Alg L重合.
Spaces of operators that are left and right modules over maximal abelian selfadjoint algebras (masa bimodules for short) are natural generalizations of algebras with commutative subspace lattices. This paper is concerned with density properties of finite rank operators and of various classes of compact operators in such modules. It is shown that every finite rank operator of a norm closed masa bimodule M is in the trace norm closure of the rank one subspace of M. An important consequence is that the rank one subspace of a strongly reflexive masa bimodule (that is, one which is the reflexive hull of its rank one operators) is dense in the module in the weak operator topology. However, in contrast to the situation for algebras, it is shown that such density need not hold in the ultraweak topology. A new method of representing masa bimodules is introduced. This uses a novel concept of anω-topology. With the appropriate notion ofω-support, a correspondence is established between reflexive masa bimodules and theirω-supports. It is shown that, if a C2-closed masa bimodule contains a trace class operator then it must contain rank one operators; indeed, every such operator is in the C2-norm closure of the rank one subspace of the module. Consequently the weak closure of any masa bimodule of trace class operators is strongly reflexive. However, the trace norm closure of the rank one subspace need not contain all trace class operators of the module. Also, it is shown that there exists a CSL algebra which contains no trace class operators yet contains an operator belonging to Cpfor allp>1. From this it follows that a transitive bimodule spanned by the rank one operators it contains need not be dense in Cpfor 1⩽p<∞. As an application, it is shown that there exists a commutative subspace lattice L such that L is non-synthetic but every weakly closed algebra which contains a masa and has invariant lattice L coincides with Alg L.