A reduction theory for operators in type ${\mathrm{I}}_{\mathrm{n}}$ von Neumann algebras

Abstract

In this paper, we study the structure of operators in a type ${\mathrm{I}}_n$ von Neumann algebra $\mathcal{A}$. Inspired by the Jordan canonical form theorem, our main motivation is to figure out the relation between the structure of an operator $A$ in $\mathcal{A}$ and the property that a bounded maximal abelian set of idempotents contained in the relative commutant $\{A{\}}^{\prime }\cap \mathcal{A}$ is unique up to similarity. Furthermore, we classify this class of operators with the property by $K$-theory for Banach algebras. Some views and techniques are from von Neumann's reduction theory.

32 pages