Celebratio Mathematica — Moore — Complete Bibliography

Complete Bibliography

Works connected to Klaus Schmidt

Filter the Bibliography List

year	title	people

			clear

C. C. Moore and K. Schmidt: “Coboundaries and homomorphisms for nonsingular actions and a problem of H. Helson,” Proc. London Math. Soc. (3) 40 : 3 (May 1980), pp. 443–475. MR 572015 Zbl 0428.28014 article

Let \( f \) be a one-cocycle for a non-singular action of a locally compact group \( G \) on a standard measure space \( (Y,\mu) \) with values in a locally compact abelian group \( A \). If \( \chi\in\hat{A} \), \( \chi(f) \) is a one-cocycle with values in the circle group \( \mathrm{\mathbf{T}} \). We investigate the question of when one can conclude that \( f \) is a coboundary, given that \( \chi(f) \) is a coboundary for some ‘sufficiently large’ set of \( \chi \). We obtain very precise results, extending previous work of Hamachi, Oka, and Osikawa. We also show that if \( G \) is measure-preserving, and \( \hat{A} \) is connected, \( f \) is a coboundary if and only if it is ‘bounded’ in a certain natural sense. We then investigate a companion problem posed by H. Helson to determine whether one can conclude that \( f \) is cohomologous to a homomorphism of \( G \) into \( A \) if the same is known to be true for all or for sufficiently many \( \chi(f) \)’s. We show that the answer is affirmative in some cases of interest but is negative in general, and, in fact, compute the defect. The counter-examples produced are examples of non-singular integer actions which have certain prescribed uncountable groups of eigenfunctions; and hence may be of some independent interest.

Klaus Schmidt	Related
Henry Berge Helson	Related