Continuity of Operator Functions in the Topology of Local Convergence in Measure

Pub Date : 2024-07-11 DOI:10.1134/s008154382401005x

A. M. Bikchentaev, O. E. Tikhonov

引用次数: 0

Abstract

Let a von Neumann algebra \(\mathcal M\) of operators act on a Hilbert space \(\mathcal{H}\), and let \(\tau\) be a faithful normal semifinite trace on \(\mathcal M\). Let \(t_{\tau\text{l}}\) be the topology of \(\tau\)-local convergence in measure on the *-algebra \(S(\mathcal M,\tau)\) of all \(\tau\)-measurable operators. We prove the \(t_{\tau\text{l}}\)-continuity of the involution on the set of all normal operators in \(S(\mathcal M,\tau)\), investigate the \(t_{\tau\text{l}}\)-continuity of operator functions on \(S(\mathcal M,\tau)\), and show that the map \(A\mapsto |A|\) is \(t_{\tau\text{l}}\)-continuous on the set of all partial isometries in \(\mathcal M\).

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测度局部收敛拓扑中算子函数的连续性

摘要让一个冯-诺依曼算子代数作用于一个希尔伯特空间，并让\(\tau\)是\(\mathcal M\) 上一个忠实的正态半无限迹。让 \(t_{\tau\text{l}}\) 是所有 \(\tau\)-measurable operators 的 *-algebra \(S(\mathcal M,\tau)\)上 \(\tau\)-local convergence in measure 的拓扑。我们证明了在\(S(\mathcal M,\tau)\中所有正常算子集合上的卷积的连续性，研究了在\(S(\mathcal M. \tau)\中算子函数的连续性、\)上的算子函数的连续性，并证明映射（A|mapsto |A|）在 \(\mathcal M\) 的所有部分等距集合上是（t_{\tau\text{l}}\）连续的。）

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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