Linear Analysis and Representation Theory

In an age when more and more items. are made to be quickly disposable or soon become obsolete due to either progress or other man caused reasons it seems almost anachronistic to write a book in the classical sense. A mathematics book becomes an indespensible companion, if it is worthy of such a relation, not by being rapidly read from cover to cover but by frequent browsing, consultation and other occasional use. While trying to create such a work I tried not to be encyclopedic but rather select only those parts of each chosen topic which I could present clearly and accurately in a formulation which is likely to last. The material I chose is all mathematics which is interesting and important both for the mathematician and to a large extent also for the mathematical physicist. I regret that at present I could not give a similar account on direct integrals and the representation theory of certain classes of Lie groups. I carefully kept the level of presentation throughout the whole book as uniform as possible. Certain introductory sections are kept shorter and are perhaps slightly more detailed in order to help the newcomer prog­ ress with it at the same rate as the more experienced person is going to proceed with his study of the details.

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.