非交换转移原理
A noncommutative transfer principle
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中文总结 AI 辅助
该文将广义Calderon转移原理推广至非交换框架，通过构造转移算子及双变量递减重排与极大算子工具，为半有限冯·诺依曼代数上的群作用建立遍历收敛结果。
中文摘要 AI 辅助
我们将文献\cite{dBL}中提出的广义Calderon转移原理推广到σ-紧局部紧Hausdorff群在 semifinite 冯·诺依曼代数$\M$上的保迹群作用的框架中。该转移原理的本质在于证明群上下文中一大类正则算子可通过该群作用转化为代数上下文中的算子。最终结果是为半有限冯·诺依曼代数的非交换Orlicz空间上的群作用建立遍历收敛结果的证明协议。我们首先证明非交换上下文中转移算子的存在性。为实际证明收敛结果，我们开发了两种工具：(1) 针对代数$L^\infty \overline{\otimes} \M$（其中$(\Gamma,\nu)$为Radon测度空间）的可称为双变量递减重排理论；(2) 适用于当前上下文的极大算子概念。随后利用这些工具将文献\cite{dBL}中的遍历收敛结果提升到非交换框架。最后，我们给出示例说明这些工具的应用。
英文摘要
We extend the generalised Calderon Transfer Principle as presented in \cite{dBL} to the setting of trace preserving group actions of $σ$-compact locally compact Hausdorff groups on semifinite von Neumann algebras $\M$. At its essence the transfer principle consists of showing that a large class of regular operators in the group context may be translated to operators in the algebra context by means of this group action. The end result is a protocol for proving ergodic convergence results for group actions on noncommutative Orlicz space of semifinite von Neumann algebras. We start with proving the existence of the transferred operator in the noncommutative context. Two tools are developed for the purpose of actually proving convergence results: (1) a theory of what may be called 2-variable decreasing rearrangements for the algebra $L^\infty \overline{\otimes} \M$ (where ($Γ,ν$) is a Radon measure space) and (2) a concepts of maximal operators suited to the present context. These tools are then used to lift the ergodic convergence results presented in \cite{dBL} to the noncommutative setting. In closing we present examples illustrating the application of the tools
发表机构
- DSI-NRF CoE in Math. and Stat. Sci, Pure and Applied Analytics NWU（南非科学与技术部-国家研究基金会数学与统计科学卓越中心，纯与应用分析西北大学）
- School of Math. & Stat. Sci., NWU（西北大学数学与统计科学学院）
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