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book and to the publisher NOORDHOFF who made possible the appearance of the second edition and enabled the author to introduce the above-mentioned modifi cations and additions. Moscow M. A. NAIMARK August 1963 FOREWORD TO THE SECOND SOVIET EDITION In this second edition the initial text has been worked over again and improved, certain portions have been completely rewritten; in particular, Chapter VIII has been rewritten in a more accessible form. The changes and extensions made by the author in the Japanese, German, first and second (= first revised) American, and also in the Romanian (lithographed) editions, were hereby taken into account. Appendices II and III, which are necessary for understanding Chapter VIII, have been included for the convenience of the reader. The book discusses many new theoretical results which have been developing in tensively during the decade after the publication of the first edition. Of course, lim itations on the volume of the book obliged the author to make a tough selection and in many cases to limit himself to simply a formulation of the new results or to pointing out the literature. The author was also compelled to make a choice of the exceptionally extensive collection of new works in extending the literature list. Monographs and survey articles on special topics of the theory which have been published during the past decade have been included in this list and in the litera ture pointed out in the individual chapters.

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Title:Normed AlgebrasFormat:PaperbackPublished:December 8, 2011Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9400992629

ISBN - 13:9789400992627

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Table of Contents

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.