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Riggings of locally compact Abelian groups. (English) Zbl 1153.22005

In this paper the authors present a construction which associates with each unitary representation of a locally compact Abelian topological group an equivalent representation and a rigged Hilbert space such that each of the unitary operators of the representation admits a generalized spectral decomposition in terms of generalized eigenvectors of them. The respective eigenvectors of the decomposition are labeled by the group characters only and the corresponding eigenvalues, which are complex numbers with modulus one, depend on the character and the group element. The spectral decomposition and the aforementioned rigging are due to the existence of a spectral measure space. The paper ends with a comment regarding the extension of the formalism in the case of nonabelian compact groups.

MSC:

22B05 General properties and structure of LCA groups
47A10 Spectrum, resolvent
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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