Natural Function Algebras - Charles E. Rickart

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Sommaire:
I. The Category of Pairs.- ? 1. Pairs and systems.- ? 2. Morphisms and extensions of pairs.- ? 3. Natural systems.- ? 4. Products of pairs.- ? 5. Examples and remarks.- II. Convexity and Naturality.- ? 6. a-convex hulls. Hull-kernel topology.- ? 7. a-convexity in a natural pair [?, a].- ? 8. Closure operations.- ? 9. Convexity and extensions.- ?10. Natural extensions.- ?11. Examples.- III. The ?ilov Boundary and Local Maximum Principle.- ?12. Independent points.- ?13. The ?ilov boundary of a pair.- ?14. A local maximum principle for natural systems.- ?15. Applications of the local maximum principle.- IV. Holomorphic Functions.- ?16. Presheaves of continuous functions.- ?17. Local extensions, ?-holomo?phic functions.- ?18. Holomorphic maps.- ?19. Examples and remarks.- V. Maximum Properties of Holomorphic Functions.- ?20. A local maximum principle for holomorphic functions.- ?21. Holomorphic peak sets.- ?22. a-presheaves.- ?23. A lemma of Glicksberg.- ?24. Maximal a-presheaves.- VI. Subharmonic Functions.- ?25. Plurisubharmonic functions in ?n.- ?26. Definitions. a-subharmonic functions.- ?27. Basic properties of a-subharmonic functions.- ?28. Plurisubharmonicity.- ?29. Maximum properties.- ?30. Integral representations.- ?31. Characterization of a-harmonic functions.- ?32. Hartog's functions.- VII. Varieties.- ?33. Varieties associated with an a-presheaf.- ?34. Convexity properties.- ?35. Generalizations of some results of Glicksberg.- ?36. Continuous families of hypersurfaces.- ?37. Remarks.- VIII. Holomorphic and Subharmonic Convexity.- ?38. Convexity with respect to an a-presheaf.- ?39. Properties of subharmonic convexity.- ?40. Naturality properties.- ?41. Holomorphic implied by subharmonic convexity.- ?42. Localproperties.- ?43. Remarks and an example.- IX. [?, a]-Domains.- ?44. Definitions.- ?45. Distance functions.- ?46. Holomorphic functions.- ?47. Relative completeness and naturality.- X. Holomorphic Extensions of [?, a]-Domains.- ?48. Morphisms and extensions. Domains of holomorphy.- ?49. Existence of maximal extensions.- ?50. Properties of maximal domains.- ?51. Remarks.- XI. Holomorphy Theory for Dual Pairs of Vector Spaces.- ?52. Generalized polynomials and holomorphic functions in a CLTS.- ?53. Dual pairs ?E, F?.- ?54. Holomorphic functions in a dual pair.- ?55. Arens holomorphic functions.- ?56. Canonical representation of dual pairs.- ?57. Derivatives.- ?58. Naturality.- XII. ?E, F? -Domains of Holomorphy.- ?59. Holomorphic functions in ?E, F?-domains.- ?60. Subdomains determined by a subspace of F.- ?61. Envelopes of holomorphy.- ?62. Series expansions.- ?63. The finite dimensional component of a domain of holomorphy.- ?64. The algebra of holomorphic functions.- ?65. Holomorphic convexity and naturality.- ?66. A Cartan-Thullen theorem.- XIII. Dual Pair Theory Applied to [?, a]-Domains.- ?67. The dual pair extension of [?, a]. A-domains.- ?68. Germ-valued functions.- ?69. Topologies for [0]?.- ?70. Naturality of algebras of germ-valued functions.- XIV. Holomorphic Extensions of ?-Domains.- ?71. Extension relative to germ-valued functions.- ?72. Uniform families of extensions.- ?73. Pseudoextensions.- ?74. Naturality properties.- Index of Symbols.- General Index.