Note on the isomorphism problem for weighted unitary operators associated with a nonsingular automorphism

K. Frączek ¹   ; M. Wysokińska ²

¹ Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland and Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland
² Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland

Colloquium Mathematicum, Tome 110 (2008) no. 1, pp. 201-204

Résumé

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.

Détail

DOI : 10.4064/cm110-1-8

Keywords: negative answer question put nadkarni ergodic conservative nonsingular automorphism widetilde mathcal widetilde consider associated unitary operators widetilde mathcal widetilde given widetilde sqrt circ cdot circ varphi cdot widetilde where varphi cocycle modulus does spectral isomorphism these operators imply varphi coboundary answer negatively example which arises infinite measure preserving transformation countable lebesgue spectrum