The similarity of irreducible operators in factors

TL;DR

This paper characterizes normal operators in separable factors that are similar to irreducible operators, advancing understanding of operator generation within von Neumann algebras.

Contribution

It provides a complete characterization of normal operators similar to irreducible operators in separable factors, linking generators and maximal abelian subalgebras.

Findings

Characterization of normal operators similar to irreducible operators

Analysis of generators related to maximal abelian subalgebras

Insights into irreducibility in separable factors

Abstract

An operator $T$ in a separable factor $\mathcal{M}$ is said to be irreducible in $\mathcal{M}$ if the von Neumann subalgebra $W^*(T)$ generated by $T$ is an irreducible subfactor of $\mathcal{M}$, i.e., $W^*(T)'\cap\mathcal{M}=\mathbb{C}I$. We say that $T$ is a single generator of $\mathcal{M}$ if $W^*(T)=\mathcal{M}$. In this paper, we study generators of separable factors related to maximal abelian self-adjoint subalgebras. As an application, we obtain a complete characterization of normal operators in separable factors which are similar to irreducible operators.