On the existence of quasicentral approximate units relative to normed ideals. Part I
On the existence of quasicentral approximate units relative to normed ideals. Part I
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DOI：
10.1016/0022-1236(90)90047-o
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发表时间：
1990-06
影响因子：
1.7
通讯作者：
D. Voiculescu
中科院分区：
文献类型：
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作者：
D. Voiculescu
令 p,, pz 为可分离希尔伯特空间上可分离 C* 代数 A 的忠实表示。# 并假设为简单起见，p,(A) 中不存在紧算子。 1211 年，我们证明了一个非交换的 Weyl-von Neumann 型定理：p 和 pZ 与紧算子取模后酉等价。 WB Arveson [2]对该定理进行了改进的阐述，并指出了拟中心近似单位在证明中的作用。研究与小于紧致的规范理想相关的摄动问题，我们在[22]中表明，非交换Weyl-von Neumann定理可以适应这种情况，主要的附加假设是相对于给定的规范理想存在准中心近似单位。对于自伴算子的交换 n 元组，我们在 [22, 231] 中表明，关于给定规范理想的扰动下的行为的一些主要问题实际上简化为相对于 n 元组的规范理想而言准中心近似单位是否存在。这包括对角性问题、绝对连续谱的守恒以及广义波算子的存在。请注意，更一般地说，通常的拟中心近似单位是 KK 理论中的重要技术成分，而用一些较小的规范理想代替紧算子为阿兰·康尼斯的非交换微分几何提供了主要背景之一。本文是 [22, 23] 的延续，包含关于准中心近似单位存在的一般事实（与换向器、无界 Fredholm 模、过滤、Macaev 理想极大性等的关系）以及具体示例的结果（非交换环面、某些离散群等）。准中心近似单位存在的障碍，例如
Let p,, pz be faithful representations of a separable C*-algebra A on a separable Hilbert space.# and assume, for simplicity, that there are no compact operators in p,(A). In 1211 we proved a non-commutative Weyl-von Neumann type theorem: p, and pZ are unitarily equivalent modulo the compact operators. WB Arveson [2] has given an improved exposition of this theorem and pointed out the role of quasicentral approximate units in the proof. Studying perturbation problems relative to normed ideals smaller than the compacts we showed in [22] that the noncommutative Weyl-von Neumann theorem can be adapted to this situation, the main additional assumption being the existence of quasicentral approximate units relative to the given normed ideal. For commuting n-tuples of self-adjoint operators we showed in [22, 231 that some of the main questions concerning the behavior under perturbations from a given normed ideal, actually reduce to the existence or non-existence of quasicentral approximate units relative to that normed ideal for the n-tuple. This includes the diagonability question, the conservation of absolutely continuous spectra and the existence of generalized wave operators. Note, more generally, that the usual quasicentral approximate units are an essential technical ingredient in KK-theory, while the replacement of the compact operators by some smaller normed ideal provides one of the main contexts for Alain Connes’ non-commutative differential geometry. The present paper is a continuation of [22, 23] and contains general facts on the existence of quasicentral approximate units (the relation to commutators, unbounded Fredholm modules, filtrations, maximality of the Macaev ideal, etc.) as well as results for specific examples (noncommutative tori, certain discrete groups, etc.). Obstructions to the existence of quasicentral approximate units, like