Author | Gaal, Steven A. author |
---|---|
Title | Linear Analysis and Representation Theory [electronic resource] / by Steven A. Gaal |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1973 |
Connect to | http://dx.doi.org/10.1007/978-3-642-80741-1 |
Descript | X, 690 p. online resource |
I. Algebras and Banach Algebras -- 1. Algebras and Norms -- 2. The Group of Units and the Quasigroup -- 3. The Maximal Ideal Space -- 4. The Spectrum of an Element -- 5. The Spectral Norm Formula -- 6. Commutative Banach Algebras and their Ideals -- 7. Radical and Semisimplicity -- 8. Involutive Algebras -- 9. H* Algebras -- Remarks -- II. Operators and Operator Algebras -- 1. Topologies on Vector Spaces and on Operator Algebras -- 2. Compact Operators -- 3. The Spectral Theorem for Compact Operators -- 4. Hilbert-Schmidt Operators -- 5. Trace Class Operators -- 6. Vector Valued Line Integrals -- 7. Homomorphisms into A. The Spectral Mapping Theorem -- 8. Unbounded Operators -- Remarks -- III. The Spectral Theorem, Stable Subspaces and v. Neumann Algebras -- 1. Linear Functionals on Vector Lattices and their Extensions -- 2. Linear Functionals on Lattices of Functions -- 3. The Spectral Theorem for SelfAdjoint Operators in Hilbert Space -- 4. Normal Elements and Normal Operators -- 5. Stable Subspaces and Commutants -- 6. von Neumann Algebras -- 7. Measures on Locally Compact Spaces -- Remarks -- IV. Elementary Representation Theory in Hilbert Space -- 1. Representations and Morphisms -- 2. Irreducible Components, Equivalence -- 3. Intertwining Operators -- 4. Schurโs Lemma -- 5. Multiplicity of Irreducible Components -- 6. The General Trace Formula -- 7. Primary Representations and Factorial v. Neumann Algebras -- 8. Algebras and Representations of Type I -- 9. Type II and III v. Neumann Algebras -- Remarks -- Preliminary Remarks to Chapter V -- V. Topological Groups, Invariant Measures, Convolutions and Representations -- 1. Topological Groups and Homogeneous Spaces -- 2. Haar Measure -- 3. Quasi-Invariant and Relatively Invariant Measures -- 4. Convolutions of Functions and Measures -- 5. The Algebra Representation Associated with ?:S??(?) -- 6. The Regular Representations of Locally Compact Groups -- 7. Continuity of Group Representations and the Gelfand-Raikov Theorem -- Remarks -- VI. Induced Representations -- 1. The Riesz-Fischer Theorem -- 2. Induced Representations when G/H has an Invariant Measure -- 3. Tensor Products -- 4. Induced Representations for Arbitrary G and H -- 5. The Existence ofa Kernel for L1(G)??(K) -- 6. The Direct Sum Decomposition of the Induced Representation ?:G?u(K) -- 7. The Isometric Isomorphism between ?2 and HS(K2, K1). The Computation of the Trace in Terms of the Associated Kernel -- 8. The Tensor Product of Induced Representations -- 9. The Theorem on Induction in Stages -- 10. Representations Induced by Representations of Conjugate Subgroups -- 11. Mackeyโs Theorem on Strong Intertwining Numbers and Some of its Consequences -- 12. Isomorphism Theorems Implying the Frobenius Reciprocity Relation -- Remarks -- VII. Square Integrable Representations, Spherical Functions and Trace Formulas -- 1. Square Integrable Representations and the Representation Theory of Compact Groups -- 2. Zonal Spherical Functions -- 3. Spherical Functions of Arbitrary Type and Height -- 4. Godementโs Theorem on the Characterization of Spherical Functions -- 5. Representations of Groups with an Iwasawa Decomposition -- 6. Trace Formulas -- Remarks -- VIII. Lie Algebras, Manifolds and Lie Groups -- 1. Lie Algebras -- 2. Finite Dimensional Representations of Lie Algebras. Cartanโs Criteria and the Theorems of Engel and Lie -- 3. Presheaves and Sheaves -- 4. Differentiable Manifolds -- 5. Lie Groups and their Lie Algebras -- 6. The Exponential Map and Canonical Coordinates -- 7. Lie Subgroups and Subalgebras -- 8. Invariant Lie Subgroups and Quotients of Lie Groups. The Projective Groups and the Lorentz Group -- Remarks -- Index of Notations and Special Symbols