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Banach algebra Borel set C+(G CC(G closed subgroup commutes compact Hausdorff space continuous function converges Corollary coset cyclic decomposition define denote dense direct integral direct sum example finite follows Fourier transform functions of positive G is compact Gelfand transform group G Haar measure Hausdorff space hence Hilbert space homomorphism identify inner product invariant inverse irreducible representations isometry isomorphism Jg/h L(S x G left Haar measure Let G Lie groups linear functional linear span Ll(G locally compact group locally compact Hausdorff Mackey measure on G Moreover multiplication neighborhood nonzero norm open set operator orbit orthogonal orthonormal basis positive type projection-valued measure Proof Proposition prove pseudomeasure quasi-invariant measure Radon measure regular representation representation of G result Schur's lemma second countable spectral theorem subalgebra subset subspace supp Suppose G system of imprimitivity theory topology trace-class trivial unique unitary equivalence unitary representation vector fields
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