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Hilbert-substructure of real measurable spaces on reductive groups. I: Basic theory. (English) Zbl 1458.43003
Summary: This paper reconsiders the age-long problem of normed linear spaces which do not admit inner product and shows that, for some subspaces, \(\mathfrak{F}_n(G)\), of real \(L^p(G)\)-spaces (when \(G\) is a reductive group in the Harish-Chandra class and \(p=2n\)), the situation may be rectified, via an outlook which generalizes the fine structure of the Hilbert space, \(L^2(G)\). This success opens the door for harmonic analysis of unitary representations, \(G\to \text{End}(\mathfrak{F}_n(G))\), of \(G\) on the Hilbert-substructure \(\mathfrak{F}_n(G)\), which has hitherto been considered impossible.
MSC:
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A80 Analysis on other specific Lie groups
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