Normed Algebras de M.A. Naimark

Normed Algebras

M.A. Naimark

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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Cuprins

I Basic Ideas from Topology and Functional Analysis.- § 1. Linear spaces.- § 2. Topological spaces.- § 3. Topological linear spaces.- § 4. Normed spaces.- § 5. Hilbert space.- § 6. Integration on locally compact spaces.- II Fundamental Concepts and Propositions in the Theory of Normed Algebras.- § 7. Fundamental algebraic concepts.- § 8. Topological algebras.- § 9. Normed algebras.- § 10. Symmetric algebras.- III Commutative Normed Algebras.- § 11. Realization of a commutative normed algebra in the form of an algebra of functions.- § 12. Homomorphism and isomorphism of commutative algebras.- § 13. Algebra (or Shilov) boundary.- § 14. Completely symmetric commutative algebras.- § 15. Regular algebras.- § 16. Completely regular commutative algebras.- IV Representations of Symmetric Algebras.- § 17. Fundamental concepts and propositions in the theory of representations.- § 18. Embedding of a symmetric algebra in an algebra of operators.- § 19. Indecomposable functionals and irreducible representations.- § 20. Application to commutative symmetric algebras.- § 21. Generalized Schur lemma.- § 22. Some representations of the algebra $$\mathfrak{B}$$(?).- V Some Special Algebras.- § 23. Completely symmetric algebras.- § 24. Completely regular algebras.- § 25. Dual algebras.- § 26. Algebras of vector-valued functions.- VI Group Algebras.- § 27. Topological groups.- § 28. Definition and fundamental properties of a group algebra.- § 29. Unitary representations of a locally compact group and their relationship with the representations of the group algebra.- § 30. Positive definite functions.- § 31. Harmonic analysis on commutative locally compact groups.- § 32. Representations of compact groups.- VII Algebras of Operators in Hilbert Space.- § 33.Various topologies in the algebra $$\mathfrak{B}$$(?).- § 34. Weakly closed subalgebras of the algebra $$\mathfrak{B}$$(?).- § 35. Relative equivalence.- § 36. Relative dimension.- § 37. Relative trace.- § 38. Structure and examples of some types of factors.- § 39. Unitary algebras and algebras with trace.- 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.- § 41. Direct integral of Hilbert spaces; the decomposition of an algebra of operators into the direct integral of irreducible algebras.- Appendix I Partially ordered sets and Zorn’s lemma.- Appendix II Borel spaces and Borel functions.- Appendix III Analytic sets.- Literature.