Linear Analysis and Representation Theory

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. Schur's 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. Mackey's 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. Godement's 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. Cartan's 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.