Normed Algebras

I Basic Ideas from Topology and Functional Analysis.- § 1. Linear spaces.- 1. Definition of a linear space.- 2. Linear dependence and independence of vectors.- 3. Subspaces.- 4. Quotient space.- 5. Linear operators.- 6. Operator calculus.- 7. Invariant subspaces.- 8. Convex sets and Minkowski functionals.- 9. Theorems on the extension of a linear functional.- § 2. Topological spaces.- 1. Definition of a topological space.- 2. Interior of a set; neighborhoods.- 3. Closed sets; closure of a set.- 4. Subspaces.- 5. Mappings of topological spaces.- 6. Compact sets.- 7. Hausdorff spaces.- 8. Normal spaces.- 9. Locally compact spaces.- 10. Stone’s theorem.- 11. Weak topology, defined by a family of functions.- 12. Topological product of spaces.- 13. Metric spaces.- 14. Compact sets in metric spaces.- 15. Topological product of metric spaces.- § 3. Topological linear spaces.- 1. Definition of a topological linear space.- 2. Closed subspaces in topological linear spaces.- 3. Convex sets in locally convex spaces.- 4. Defining a locally convex topology in terms of seminorms.- 5. The case of a finite-dimensional space.- 6. Continuous linear functionals.- 7. Conjugate space.- 8. Convex sets in a finite-dimensional space.- 9. Convex sets in the conjugate space.- 10. Cones.- 11. Annihilators in the conjugate space.- 12. Analytic vector-valued functions.- 13. Complete locally convex spaces.- § 4. Normed spaces.- 1. Definition of a normed space.- 2. Series in a normed space.- 3. Quotient spaces of a Banach space.- 4. Bounded linear operators.- 5. Bounded linear functionals; conjugate space.- 6. Compact (or completely continuous) operators.- 7. Analytic vector-valued functions in a Banach space.- § 5. Hilbert space.- 1. Definition of Hilbert space.- 2. Projection of a vector on a subspace.- 3. Bounded linear functionals in Hilbert space.- 4. Orthogonal systems of vectors in Hilbert space.- 5. Orthogonal sum of subspaces.- 6. Direct sum of Hilbert spaces.- 7. Graph of an operator.- 8. Closed operators; closure of an operator.- 9. Adjoint operator.- 10. The case of a bounded operator.- 11. Generalization to operators in a Banach space.- 12. Projection operators.- 13. Reducibility.- 14. Partially isometric operators.- 15. Matrix representation of an operator.- § 6. Integration on locally compact spaces.- 1. Fundamental concepts; formulation of the problem.- 2. Fundamental properties of the integral.- 3. Extension of the integral to lower semi-continuous functions.- 4. Upper integral of an arbitrary nonnegative real-valued function.- 5. Exterior measure of a set.- 6. Equivalent functions.- 7. The spaces ?1 and L1.- 8. Summable sets.- 9. Measurable sets.- 10. Measurable functions.- 11. The real space L2.- 12. The complex space L2.- 13. The space L?.- 14. The positive and negative parts of a linear functional.- 15. The Radon-Nikodým theorem.- 16. The space conjugate to L1.- 17. Complex measures.- 18. Integrals on the direct product of spaces.- 19. The integration of vector-valued and operator-valued functions.- II Fundamental Concepts and Propositions in the Theory of Normed Algebras.- § 7. Fundamental algebraic concepts.- 1. Definition of a linear algebra.- 2. Algebras with identity.- 3. Center.- 4. Ideals.- 5. The (Jacobson) radical.- 6. Homomorphism and isomorphism of algebras.- 7. Regular representations of algebras.- § 8. Topological algebras.- 1. Definition of a topological algebra.- 2. Topological adjunction of the identity.- 3. Algebras with continuous inverse.- 4. Resolvents in an algebra with continuous inverse.- 5. Topological division algebras with continuous inverse.- 6. Algebras with continuous quasi-inverse.- § 9. Normed algebras.- 1. Definition of a normed algebra.- 2. Adjunction of the identity.- 3. The radical in a normed algebra.- 4. Banach algebras with identity.- 5. Resolvent in a Banach algebra with identity.- 6. Continuous homomorphisms of normed algebras.- 7. Regular representations of a normed algebra.- § 10. Symmetric algebras.- 1. Definition and simplest properties of a symmetric algebra.- 2. Positive functional.- 3. Normed symmetric algebras.- 4. Positive functional in a symmetric Banach algebra.- III Commutative Normed Algebras.- § 11. Realization of a commutative normed algebra in the form of an algebra of functions.- 1. Quotient algebra modulo a maximal ideal.- 2. Functions on maximal ideals generated by elements of an algebra.- 3. Topologization of the set of all maximal ideals.- 4. The case of an algebra without identity.- 5. System of generators of an algebra.- 6. Analytic functions of algebra elements.- 7. Wiener pairs of algebras.- 8. Functions of several algebra elements; locally analytic functions.- 9. Decomposition of an algebra into the direct sum of ideals.- 10. Algebras with radical.- § 12. Homomorphism and isomorphism of commutative algebras.- 1. Uniqueness of the norm in a semisimple algebra.- 2. The case of symmetric algebras.- § 13. Algebra (or Shilov) boundary.- 1. Definition and fundamental properties of the algebra boundary.- 2. Extension of maximal ideals.- § 14. Completely symmetric commutative algebras.- 1. Definition of a completely symmetric algebra.- 2. Criterion for complete symmetry.- 3. Application of Stone’s theorem.- 4. The algebra boundary of a completely symmetric algebra.- § 15. Regular algebras.- 1. Definition of a regular algebra.- 2. Normal algebras of functions.- 3. Structure space of an algebra.- 4. Properties of regular algebras.- 5. The case of an algebra without identity.- 6. Sufficient condition that an algebra be regular.- 7. Primary ideals.- § 16. Completely regular commutative algebras.- 1. Definition and simplest properties of a completely regular algebra.- 2. Realization of completely regular commutative algebras.- 3. Generalization to multi-normed algebras.- 4. Symmetric subalgebras of the algebra C(T) and compact extensions of the space T.- 5. Antisymmetric subalgebras of the algebra C(T).- 6. Subalgebras of the algebra C(T) and certain problems in approximation theory.- IV Representations of Symmetric Algebras.- § 17. Fundamental concepts and propositions in the theory of representations.- 1. Definitions and simplest properties of a representation.- 2. Direct sum of representations.- 3. Description of representations in terms of positive functionals.- 4. Representations of completely regular commutative algebras; spectral theorem.- 5. Spectral operators.- 6. Irreducible representations.- 7. Connection between vectors and positive functionals.- § 18. Embedding of a symmetric algebra in an algebra of operators.- 1. Regular norm.- 2. Reduced algebras.- 3. Minimal regular norm.- § 19. Indecomposable functionals and irreducible representations.- 1. Positive functionals, dominated by a given positive functional.- 2. The algebra Cf.- 3. Indecomposable positive functionals.- 4. Completeness and approximation theorems.- § 20. Application to commutative symmetric algebras.- 1. Minimal regular norm in a commutative symmetric algebra.- 2. Positive functionals in a commutative symmetric algebra.- 3. Examples.- 4. The case of a completely symmetric algebra.- § 21. Generalized Schur lemma.- 1. Canonical decomposition of an operator.- 2. Fundamental theorem.- 3. Application to direct sums of pairwise non-equivalent representations.- 4. Application to representations which are multiples of a given irreducible representation.- § 22. Some representations of the algebra $$\mathfrak{B}$$(?).- 1. Ideals in the algebra $$\mathfrak{B}$$(?).- 2. The algebra I0 and its representations.- 3. Representations of the algebra $$\mathfrak{B}$$(?).- V Some Special Algebras.- § 23. Completely symmetric algebras.- 1. Definition and examples of completely symmetric algebras.- 2. Spectrum.- 3. Theorems on extensions.- 4. Criterion for complete symmetry.- § 24. Completely regular algebras.- 1. Fundamental properties of completely regular algebras.- 2. Realization of a completely regular algebra as an algebra of operators.- 3. Quotient algebra of a completely regular algebra.- § 25. Dual algebras.- 1. Annihilator algebras and dual algebras.- 2. Ideals in an annihilator algebra.- 3. Semisimple annihilator algebras.- 4. Simple annihilator algebras.- 5. H*-algebras.- 6. Completely regular dual algebras.- § 26. Algebras of vector-valued functions.- 1. Definition of an algebra of vector-valued functions.- 2. Ideals in an algebra of vector-valued functions.- 3. Conditions for a vector-valued function to belong to an algebra.- 4. The case of completely regular algebras.- 5. Continuous analogue of the Schur lemma.- 6. Structure space of a completely regular algebra.- VI Group Algebras.- § 27. Topological groups.- 1. Definition of a group.- 2. Subgroups.- 3. Definition and simplest properties of a topological group.- 4. Invariant integrals and invariant measures on a locally compact group.- 5. Existence of an invariant integral on a locally compact group.- § 28. Definition and fundamental properties of a group algebra.- 1. Definition of a group algebra.- 2. Some properties of the group algebra.- § 29. Unitary representations of a locally compact group and their relationship with the representations of the group algebra.- 1. Unitary representations of a group.- 2. Relationship between representations of a group and of the group algebra.- 3. Completeness theorem.- 4. Examples.- a) Unitary representations of the group of linear transformations of the real line.- b) Unitary representations of the proper Lorentz group.- c) Example of a group algebra which is not completely symmetric.- § 30. Positive definite functions.- 1. Positive definite functions and their relationship with unitary representations.- 2. Relationship of positive definite functions with positive functional on a group algebra.- 3. Regular sets.- 4. Trigonometric polynomials on a group.- 5. Spectrum.- § 31. Harmonic analysis on commutative locally compact groups.- 1. Maximal ideals of the group algebra of a commutative group; characters.- 2. Group of characters.- 3. Positive definite functions on a commutative group.- 4. Inversion formula and Plancherel’s theorem for commutative groups.- 5. Separation property of the set [L1 ? P].- 6. Duality theorem.- 7. Unitary representations of commutative groups.- 8. Theorems of Tauberian type.- 9. The case of a compact group.- 10. Spherical functions.- 11. The generalized translation operation.- § 32. Representations of compact groups.- 1. The algebra L2($$\mathfrak{G}$$).- 2. Representations of a compact group.- 3. Tensor product of representations.- 4. Duality theorem for a compact group.- VII Algebras of Operators in Hilbert Space.- § 33. Various topologies in the algebra $$\mathfrak{B}$$(?).- 1. Weak topology.- 2. Strong topology.- 3. Strongest topology.- 4. Uniform topology.- § 34. Weakly closed subalgebras of the algebra $$\mathfrak{B}$$(?).- 1. Fundamental concepts.- 2. Principal identity.- 3. Center.- 4. Factorization.- § 35. Relative equivalence.- 1. Operators and subspaces adjoined to an algebra.- 2. Fundamental lemma.- 3. Definition of relative equivalence.- 4. Comparison of closed subspaces.- 5. Finite and infinite subspaces.- § 36. Relative dimension.- 1. Entire part of the ratio of two subspaces.- 2. The case when a minimal subspace exists.- 3. The case when a minimal subspace does not exist.- 4. Existence and properties of relative dimension.- 5. The range of the relative dimension; classification of factors.- 6. Invariance of factor type under symmetric isomorphisms.- § 37. Relative trace.- 1. Definition of trace.- 2. Properties of the trace.- 3. Traces in factors of types (I?) and (II?).- § 38. Structure and examples of some types of factors.- 1. The mapping M ? M($$\mathfrak{M}$$).- 2. Matrix description of factors of types (I) and (II).- 3. Description of factors of type (I).- 4. Structure of factors of type (II?).- 5. Example of a factor of type (II1).- 6. Approximately finite factors of type (II1).- 7. Relationship between the types of factors M and M?.- 8. Relationship between symmetric and spatial isomorphisms.- 9. Unbounded operators, adjoined to a factor of finite type.- § 39. Unitary algebras and algebras with trace.- 1. Definition of a unitary algebra.- 2. Definition of an algebra with trace.- 3. Unitary algebras defined by the trace.- 4. Canonical trace in a unitary algebra.- VIII Decomposition of an Algebra of Operators into Irreducible Algebras.- § 40. Formulation of the problem; canonical form of a commutative algebra of operators in Hilbert space.- 1. Formulation of the problem.- 2. The separability lemma.- 3. Canonical form of a commutative algebra.- § 41. Direct integral of Hilbert spaces; the decomposition of an algebra of operators into the direct integral of irreducible algebras.- 1. Direct integral of Hilbert spaces.- 2. Decomposition of a Hilbert space into a direct integral with respect to a given commutative algebra R.- 3. Decomposition with respect to a maximal commutative algebra; condition for irreducibility.- 4. Decomposition of a unitary representation of a locally compact group into irreducible representations.- 5. Central decompositions and factor representations.- 6. Representations in a space with an indefinite metric.- Appendix I Partially ordered sets and Zorn’s lemma.- Appendix II Borel spaces and Borel functions.- Appendix III Analytic sets.- Literature.