关于von Neumann代数中几类投影运算的研究

Studies of Several Projection Operators in von Neumann Algebras

摘要: 本文研究了在von Neumann代数框架下，投影运算后等价的保持情形，特别是在不同类型的von Neumann代数中投影等价的刻画及其性质。设$E$ 、$F$ 、$N$ 为von Neumann代数$ℳ$ 中的三个有限投影，且$E~F$ 。当$NE=NF=0$ 时，那么有$N+E~N+F$ ，$N\wedge E~N\wedge F$ ，$N\vee E~N\vee F$ 成立。当 $N<E,N<F$ 时，那么有$N\wedge E~N\wedge F$ ，$N\vee E~N\vee F$ 成立。当$E<N,F<N$ 时，那么有 $N-E~N-F$ ，$N\wedge E~N\wedge F$ ，$N\vee E~N\vee F$ 成立。本文进一步推至$N$ 为von Neumann代数$ℳ$ 中的无限投影，并且考虑了von Neumann代数$ℳ$ 中四个投影运算的情况。

Abstract: This paper studies the conditions of equivalents after projection operations under the framework of von Neumann algebra, especially the characterization and properties of projection equivalents in different types of von Neumann algebras. Let $E,F,N$ be three finite projections in von Neumann algebra $ℳ$ , and $E~F$ . If $NE=NF=0$ , then it follows that $N+E~N+F$ , $N\wedge E~N\wedge F$ , $N\vee E~N\vee F$ holds. If $N<E$ , $N<F$ , then it follows that $N\wedge E~N\wedge F$ , $N\vee E~N\vee F$ holds. If $E<N$ , $F<N$ , then it follows that $N-E~N-F$ , $N\wedge E~N\wedge F$ , $N\vee E~N\vee F$ holds. This paper further extends these results to infinite projections $N$ in von Neumann algebras $ℳ$ and considers the case of four projection operations within von Neumann algebras $ℳ$ .