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Journal of the London Mathematical Society 1990 s2-42(2):329-338; doi:10.1112/jlms/s2-42.2.329
© 1990 by London Mathematical Society
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© Oxford University Press

Spectral Extension for Measures

Bernard Host and François Parreau

Département de Mathématiques et Informatique, CSP, Université Paris-Nord Avenue J.-B. Clément, 93430 Villetaneuse, France

The main difficulty when working on spectral properties of measures on a locally compact abelian group lies in the fact that the algebra M(G) has a very large and intricate Gelfand spectrum, even when the group G is T or R. Besides, every measure or finite collection of measures in M{G) is contained in a small subalgebra isomorphic to some space Ll(µ), and the characters (multiplicative linear functionals) of such a subalgebra are the solutions of a simple functional equation in L{infty}(µ). The natural question is which of these characters can be extended to M{G). We give an explicit criterion based on a point property and a more theoretical condition which shows that spectral properties of measures can be determined inside such small subalgebras.


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