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

TL;DR

This paper develops a reduction theory for operators in type Iₙ von Neumann algebras, linking their structure to maximal abelian sets of idempotents and classifying them via K-theory, inspired by Jordan canonical form and von Neumann's reduction.

Contribution

It introduces a new reduction framework for operators in type Iₙ von Neumann algebras and classifies these operators using K-theory, connecting algebraic structure with operator properties.

Findings

Characterization of operators via maximal abelian sets of idempotents

Uniqueness of these sets up to similarity

Classification of operators using K-theory

Abstract

In this paper, we study the structure of operators in a type $\mathrm{I}_{n}$ von Neumann algebra $\mathscr{A}$. Inspired by the Jordan canonical form theorem, our main motivation is to figure out the relation between the structure of an operator $A$ in $\mathscr{A}$ and the property that a bounded maximal abelian set of idempotents contained in the relative commutant $\{A\}^{\prime} \cap\mathscr{A}$ 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.