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arXiv math/0509691math.OA
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von Neumann algebras中正规算子的酉轨道

Unitary orbits of normal operators in von Neumann algebras

David Sherman

摘要:

本文的出发点是vonNeumann代数中正规算子酉轨道的范数和强 * 闭包的简单描述。这些陈述是根据光谱数据,不依赖于代数的类型或基数。我们将这与几个已知的结果,并得出一些后果,其中我们列出了一些在这里。当环境von Neumann代数是σ-有限代数的直和时,任意两个正规算子具有相同的范数闭酉轨道当且仅当它们具有相同的强 *-闭酉轨道当且仅当它们具有相同的强-闭酉轨道.但是这三个闭包通常彼此不同,并且与未闭合的酉轨道不同,并且我们描述了这四个集合中的任何两个之间何时相等。我们还表明,在一个真正的无限冯诺依曼代数,强封闭酉轨道的任何运营商,不一定正常,满足中心(非空)的左本质中心谱的Halpern。一个推论是“III型Weyl-von Neumann-Berg定理”,涉及本质中心谱的包容性。

英文摘要:

The starting points for this paper are simple descriptions of the norm and strong* closures of the unitary orbit of a normal operator in a von Neumann algebra. The statements are in terms of spectral data and do not depend on the type or cardinality of the algebra. We relate this to several known results and derive some consequences, of which we list a few here. Exactly when the ambient von Neumann algebra is a direct sum of sigma-finite algebras, any two normal operators have the same norm-closed unitary orbit if and only if they have the same strong*-closed unitary orbit if and only if they have the same strong-closed unitary orbit. But these three closures generally differ from each other and from the unclosed unitary orbit, and we characterize when equality holds between any two of these four sets. We also show that in a properly infinite von Neumann algebra, the strong-closed unitary orbit of any operator, not necessarily normal, meets the center in the (non-void) left essential central spectrum of Halpern. One corollary is a "type III Weyl-von Neumann-Berg theorem" involving containment of essential central spectra.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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