Calkin算子空间上的保不交映射与正等距映射
Disjointness-preserving mappings on Calkin operator spaces and positive isometries
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中文总结 AI 辅助
该研究在半有限冯·诺依曼代数框架下，证明了保不交映射恒为零的条件，给出正等距映射保不交的结论，得到其一般形式，推广并统一了相关经典结果。
中文摘要 AI 辅助
设$E(\tilde{\tau})$和$F(\tilde{\tau})$是两个依附于半有限冯·诺依曼代数$\tilde{\tau}$的Calkin算子空间，其中$\tilde{\tau}$配备了半有限忠实正规迹$\tau$。我们证明：若$\tilde{\tau}$是无原子的，$\tau$是有限的，且$E(\tilde{\tau})\not\tilde{\tau}F(\tilde{\tau})$，则每个序-测度连续且保不交的映射$T:E(\tilde{\tau})\to F(\tilde{\tau})$都恒为零映射，这建立了Abramovich定理的非交换版本。我们还证明：若$F(\tilde{\tau})$的范数是严格单调的，则每个从$\tau$可测算子的赋范$\tilde{\tau}$-双模$E(\tilde{\tau})$到另一$\tau$可测算子的赋范$\tilde{\tau}$-双模$F(\tilde{\tau})$的正等距映射$T$都保不交。作为应用，我们得到了$T$的一般形式，该形式推广并统一了Abramovich、de Jager、Conradie、Veksler和Sukochev等人的若干结果\ucb1991,vek,dC20。
英文摘要
Let $E(\mathcal{M},τ)$ and $F(\mathcal{M},τ)$ be two Calkin operator spaces affiliated with a semifinite von Neumann algebra $\mathcal{M}$ equipped with a semifinite faithful normal trace $τ$. We show that if $\mathcal{M}$ is atomless, $τ$ is finite, and $E(v,τ)\not\subseteq F(\mathcal{M},τ)$, then every order-measure continuous and disjointness-preserving mapping $T:E(\mathcal{M},τ)\xrightarrow{\rm into} F(\mathcal{M},τ)$ is identical to the zero mapping, which establishes a noncommutative version of Abramovich's theorem. We also show that every positive isometry $T$ from a normed $\mathcal{M}$-bimodule $E(\mathcal{M},τ)$ of $τ$-measurable operators into another $F(\mathcal{M},τ)$ preserves disjointness provided that the norm of $F(\mathcal{M},τ)$ is strictly monotone. As an application, we obtain the general form of $T$, which extends and unifies several results due to Abramovich, de Jager, Conradie, Veksler and Sukochev et al. \cite{SV,HSZ20,Abra1991,vek,dC20}.
