关于Von Neumann代数中单性投影的研究

Studies of Monic Projections in Von Neumann Algebras

摘要: von Neumann代数是算子代数中的重要组成部分，而von Neumann代数都可以由它的投影生成，因此投影就成为了算子代数中一类重要的研究对象。本文研究了有限von Neumann代数中的单性投影，并对它的性质进行了一些刻画。设$ℳ$ 是作用于Hilbert空间$\mathscr{H}$ 上的$I_n$ 型von Neumann代数，$\mathcal{A}$ 是$ℳ$ 的极大交换子代数，投影$E$ 、$F$ 是$\mathcal{A}$ 中的两个单性投影，$\left\{E_1,\cdot \cdot \cdot ,E_n\right\}$ 与$\left\{F_1,\cdot \cdot \cdot ,F_n\right\}$ 为$ℳ$ 中的两列相互正交的投影族，并有${\displaystyle \sum _{i=1}^n{E}_i}=I$ ，${\displaystyle \sum _{i=1}^n{F}_i}=I$ ，$E\sim E_i$ 与$F\sim F_i$ 成立，这里$1\le i\le n$ 为正整数，那么对任意的$\epsilon >0$ ，存在一个超弱连续正线性映射$\tau :ℳ\to \mathcal{C}$ ，使得$\tau \left(E\right)\le \left(1+\epsilon \right)\tau \left(F\right)$ 成立。

Abstract: Von Neumann algebras are an important part of operator algebras, and von Neumann algebras can be generated by its projections, so projections are an important research topic in operator algebras. In this paper, the monic projections in finite von Neumann algebras are studied, and some characterizations of their properties are given. If $ℳ$ is a $I_n$ type von Neumann algebra acting on a Hilbert space $\mathscr{H}$ , and let $\mathcal{A}$ be a maximal abelian subalgebra of $ℳ$ , and $E$ , $F$ be two monic projections in $\mathcal{A}$ , $\left\{E_1,\cdot \cdot \cdot ,E_n\right\}$ and $\left\{F_1,\cdot \cdot \cdot ,F_n\right\}$ are two mutually orthogonal projections families, and ${\displaystyle \sum _{i=1}^n{E}_i}=I$ , ${\displaystyle \sum _{i=1}^n{F}_i}=I$ , $E\sim E_i$ and $F\sim F_i$ , where $i\in N^{+}$ , then for any $\epsilon >0$ , there is an ultraweakly continuous positive linear mapping $\tau :ℳ\to \mathcal{C}$ such that $\tau \left(E\right)\le \left(1+\epsilon \right)\tau \left(F\right)$ .