TY - JOUR
AU - Zhu, Zhangsheng
AU - Fang, Junsheng
AU - Shi, Rui
PY - 2017/05/27
Y2 - 2022/01/22
TI - On a class of operators in the hyperfinite $\mathrm{II}_1$ factor
JF - MATHEMATICA SCANDINAVICA
JA - Math. Scand.
VL - 120
IS - 2
SE - Articles
DO - 10.7146/math.scand.a-25625
UR - https://www.mscand.dk/article/view/25625
SP - 249-271
AB - <p>Let $R$ be the hyperfinite $\mathrm {II}_1$ 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.</p>
ER -