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Spectral Extension for Measures
Author(s) -
Host Bernard,
Parreau François
Publication year - 1990
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-42.2.329
Subject(s) - subalgebra , multiplicative function , mathematics , abelian group , extension (predicate logic) , pure mathematics , locally compact space , measure (data warehouse) , spectrum (functional analysis) , space (punctuation) , simple (philosophy) , group (periodic table) , property (philosophy) , point (geometry) , discrete mathematics , algebra over a field , mathematical analysis , computer science , physics , programming language , philosophy , geometry , epistemology , quantum mechanics , database , operating system
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 L l (μ), and the characters (multiplicative linear functionals) of such a subalgebra are the solutions of a simple functional equation in L ∞ (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.