A generalization of Voiculescu's theorem for normal operators to semifinite von Neumann algebras
A generalization of Voiculescu's theorem for normal operators to semifinite von Neumann algebras
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将正规算子的 Voiculescu 定理推广到半有限冯·诺依曼代数
DOI：
10.1016/j.aim.2020.107347
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发表时间：
2020
影响因子：
1.7
通讯作者：
Rui Shi
中科院分区：
文献类型：
--
作者：
Qihui Li;Junhao Shen;Rui Shi
本文给出了Voiculescu正规算子定理的一个推广形式，证明了在具有可分预对偶和忠实正规半有限迹权τ的von Neumann代数中，每个正规算子都是对角算子的任意小（max，2}）-范数扰动.进一步，在具有忠实正规半有限迹权的可数可分解真无限von Neumann代数中，证明了每个自伴算子都可以对角化为满足一定自然条件的模赋范理想.在半有限因子的情况下，对核C-代数，Voiculescu的吸收定理[33，定理2.4]也得到了类似的证明。
In this paper, we provide a generalized version of Voiculescu's theorem for normal operators by showing that, in a von Neumann algebra with separable pre-dual and a faithful, normal, semifinite, tracial weight τ, each normal operator is an arbitrarily small (max{‖⋅‖,‖⋅‖ 2})-norm perturbation of a diagonal operator. Furthermore, in a countably decomposable, properly infinite von Neumann algebra with a faithful normal semifinite tracial weight, we prove that each self-adjoint operator can be diagonalized modulo normed ideals satisfying a natural condition. An analogue, for nuclear C⁎-algebras, of Voiculescu's absorption theorem [33, Theorem 2.4] is also proved in the case of semifinite factors.