abelian adjoint arbitrary associated Banach algebra Banach space belongs called choose closed subgroup commutative compact set compact support complex Banach contains continuous functions continuous linear functional Corollary defined definition denote differentiable direct sum element equivalence class exists finite dimensional follows given group G Haar measure Hence Hilbert space homomorphism HS operator idempotent implies induced representations inner product integral invariant measure invariant subspace involutive algebra irreducible components irreducible representation isometry isomorphism kernel left ideal left invariant Lemma Let G Lie algebra Lie group linear manifold linear operator locally compact group matrix minimal MONS morphism Neumann algebra nilpotent non-zero norm obtain open set orthogonal projection Proof Proposition 9 prove regular representation respect restriction result satisfying Section semisimple sheaf shows Similarly solvable strongly continuous structure subalgebra subgroup of G subset suppose Theorem tion topological group unimodular unitary representation zero zonal spherical function

 

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