Note on the isomorphism problem for weighted unitary operators associated with a nonsingular automorphism
Volume 110 / 2008
Colloquium Mathematicum 110 (2008), 201-204
MSC: 37A40, 37A30.
DOI: 10.4064/cm110-1-8
Abstract
We give a negative answer to a question put by Nadkarni: Let $S$ be an ergodic, conservative and nonsingular automorphism on $(\widetilde{X},\mathcal{B}_{\widetilde{X}},m)$. Consider the associated unitary operators on $L^2(\widetilde{X},\mathcal{B}_{\widetilde{X}},m)$ given by $\widetilde{U}_Sf=\sqrt{{d(m\circ S)}/{dm}}\cdot (f\circ S)$ and $\varphi\cdot \widetilde{U}_S$, where $\varphi$ is a cocycle of modulus one. Does spectral isomorphism of these two operators imply that $\varphi$ is a coboundary? To answer it negatively, we give an example which arises from an infinite measure-preserving transformation with countable Lebesgue spectrum.
