On a class of operators in the hyperfinite ${\mathrm{I}\mathrm{I}}_1$ factor

Let $R$ be the hyperfinite ${\mathrm{I}\mathrm{I}}_1$ factor and let $u,v$ be two generators of $R$ such that $u^{\ast }u=v^{\ast }v=1$ 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 $C^{\ast }$-algebra generated by $uf(v)$ and the identity operator is a generalized universal irrational rotation $C^{\ast }$-algebra.