Functional analytic aspects of non-commutative harmonic analysis

Given a homogeneous space X = G/H with an invariant measure it is shown, using Grothendieck's inequality, that a G-invariant Hilbert subspace of the space of distributions of order zero on X is actually contained in Lloc2(X). Moreover, if θ is an automorphism on G appropriately related to H, it is shown that, under condition that H-orbits are smooth, an H-bi-invariant distribution of positive type on G satisfies the identity Ťθ = T if the corresponding Hilbert space is contained in Lloc2(X). This shows that, under the smooth orbit condition, G-invariant Hilbert subspaces of Lloc2 (X) have a unique decomposition into irreducible Hilbert spaces as in the case of generalized Gelfand pairs.