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Coboundaries and Homomorphisms for Non‐Singular Actions and a Problem of H. Helson
Author(s) -
Moore Calvin C.,
Schmidt Klaus
Publication year - 1980
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-40.3.443
Subject(s) - mathematics , uncountable set , abelian group , homomorphism , bounded function , group (periodic table) , group action , measure (data warehouse) , space (punctuation) , locally compact space , pure mathematics , combinatorics , action (physics) , discrete mathematics , mathematical analysis , linguistics , chemistry , philosophy , physics , countable set , organic chemistry , quantum mechanics , database , computer science
Let f be a one‐cocycle for a non‐singular action of a locally compact group G on a standard measure space ( Y , μ) with values in a locally compact abelian group A . If χ ɛ Â, χ( f ) is a one‐cocycle with values in the circle group T . We investigate the question of when one can conclude that f is a coboundary, given that χ( f ) is a coboundary for some ‘sufficiently large’ set of χ. We obtain very precise results, extending previous work of Hamachi, Oka, and Osikawa. We also show that if G is measure‐preserving, and  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 χ( 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.