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

K. Frączek; M. Wysokińska

Colloquium Mathematicae, Tome 111 (2008), p. 201-204 / Harvested from The Polish Digital Mathematics Library

Résumé

We give a negative answer to a question put by Nadkarni: Let S be an ergodic, conservative and nonsingular automorphism on $(X ̃,_{X ̃}, m)$ . Consider the associated unitary operators on $L ² (X ̃,_{X ̃}, m)$ given by $U ̃_{S} f = \sqrt (d (m \circ S) / d m) \cdot (f \circ S)$ and $φ \cdot U ̃_{S}$ , where φ is a cocycle of modulus one. Does spectral isomorphism of these two operators imply that φ is a coboundary? To answer it negatively, we give an example which arises from an infinite measure-preserving transformation with countable Lebesgue spectrum.

Publié le : 2008-01-01

Zbl 1142.37009

EUDML-ID : urn:eudml:doc:284013

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     title = {Note on the isomorphism problem for weighted unitary operators associated with a nonsingular automorphism},
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     year = {2008},
     pages = {201-204},
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K. Frączek; M. Wysokińska. Note on the isomorphism problem for weighted unitary operators associated with a nonsingular automorphism. Colloquium Mathematicae, Tome 111 (2008) pp. 201-204. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm110-1-8/