von Neumann代数上保持绝对连续和奇异的映射
Maps Preserving Absolute Continuity and Singularity on a von Neumann Algebra
DOI: 10.12677/AAM.2021.105177, PDF,    国家自然科学基金支持
作者: 车晶晶, 刘爱芳：太原理工大学数学学院，山西 晋中
关键词: von Neumann代数；线性保持；绝对连续；奇异；von Neumann Algebra； Linear Preserver； Absolute Continuity； Singular
摘要: 设A是无限维复Hilbert空间上的一个von Neumann代数。A+为所有正算子的锥。本文证明了一个双射φ:A+→A+在两个方向上都保持绝对连续，则其在两个方向上也保持奇异。并证明了这个双射φ可以由有界、可逆、线性或共轭线性的算子来刻画。
Abstract: Let A be a von Neumann algebra on an infinite dimensional, complex Hilbert space. A+ stands for the cone of all positive operators. In this paper, we obtain that bijective maps φ:A+→A+ that preserves absolute continuity in both directions are also preserve singularity in both directions. Moreover, we show that these maps φ can be characterized by invertible, linear or conjugate linear operators.
文章引用：车晶晶, 刘爱芳. von Neumann代数上保持绝对连续和奇异的映射[J]. 应用数学进展, 2021, 10(5): 1661-1667. https://doi.org/10.12677/AAM.2021.105177

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