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" Normed Algebras "
by M.A. Naimark.
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BL
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774095
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b594089
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| Main Entry
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by M.A. Naimark.
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Normed Algebras\ by M.A. Naimark.
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| Publication Statement
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Dordrecht : Springer Netherlands, 1979
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| ISBN
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9400992602
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: 9400992629
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: 9789400992603
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: 9789400992627
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| Contents
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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-Nikodym 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
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| Subject
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Mathematics.
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| LC Classification
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QA326.B963 1979
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| Added Entry
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M A Naimark
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