On a class of operators in the hyperfinite ${\rm II}_1$ factor

TL;DR

This paper investigates operators of the form $uf(v)$ in the hyperfinite ${ m II}_1$ factor, calculating their spectra, Brown spectra, and exploring their invariant subspaces, revealing their algebraic structure and subfactor properties.

Contribution

It introduces a detailed analysis of the spectral properties and subfactor structure of operators $uf(v)$ in the hyperfinite ${ m II}_1$ factor, including their generated algebras.

Findings

Spectrum and Brown spectrum of $uf(v)$ are explicitly calculated.

The von Neumann algebra generated by $uf(v)$ is an irreducible subfactor with finite index.

The $C^*$-algebra generated by $uf(v)$ is a generalized universal irrational rotation algebra.

Abstract

Let $R$ be the hyperfinite ${\rm II}_1$ factor and let $u,v$ be two generators of $R$ such that $u^*u=v^*v=1$ and $vu=e^{2\pi i\theta} uv$ for an irrational number $\theta$. In this paper we study the class of operators $uf(v)$, where $f$ is a bounded Lebesgue measurable function on the unit circle $S^1$. We calculate the spectrum and Brown spectrum of operators $uf(v)$, and study the invariant subspace problem of such operators relative to $R$. We show that under general assumptions the von Neumann algebra generated by $uf(v)$ is an irreducible subfactor of $R$ with index $n$ for some natural number $n$, and the $C^*$-algebra generated by $uf(v)$ and the identity operator is a generalized universal irrational rotation $C^*$-algebra.