Linear Analysis and Representation Theory
A frequently cited scholarly text, this unified treatment constitutes an ideal reference for graduate math students and mathematicians. Its clear, accurate formulations offer a relatively self-contained resource for those studying locally compact groups, Lie groups and algebras, harmonic analysis, and the representation of groups.
An introductory chapter on algebras and Banach algebras is followed by a summary of linear functional analysis and examinations of spectral theorem, stable subspaces, and von Neumann algebras. Subsequent chapters explore elementary representation theory in Hilbert space; topological groups, invariant measures, convolutions, and representations; the basic theory of induced representations; and fundamentals of the theory of representations of compact groups. The final chapters offer a concise introduction to the theory of Lie algebras, Lie groups, and transform groups.
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JA Chapter I Algebras and Banach Algebras Algebras and Norms
The Group of Units and the Quasigroup
The Maximal Ideal Space
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abelian adjoint arbitrary associated Banach algebra Banach space belongs called choose closed subgroup commutative compact set compact support contains continuous functions continuous linear functional Corollary deﬁned denote differentiable direct sum element equivalence class exists ﬁnd ﬁnite dimensional ﬁrst ﬁxed follows function f given group G Haar measure hand side Hence Hilbert space homomorphism HS operator idempotent implies induced representations inﬁnite inner product integral invariant measure invariant subspace involutive algebra irreducible components irreducible representation isometry isomorphism kernel left ideal left invariant Lemma Let 9 let f 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 regular representation restriction result satisﬁes satisfying Section semisimple sheaf shows Similarly solvable structure subalgebra subgroup of G suppose Theorem tion topological group unimodular unitary representation vector ﬁeld weakly zero zonal spherical function