Natural Function Algebras

The term "function algebra" usually refers to a uniformly closed algebra of complex valued continuous functions on a compact Hausdorff space. Such Banach alge bras, which are also called "uniform algebras", have been much studied during the past 15 or 20 years. Since the most important examples of uniform algebras consist of, or are built up from, analytic functions, it is not surprising that most of the work has been dominated by questions of analyticity in one form or another. In fact, the study of these special algebras and their generalizations accounts for the bulk of the re search on function algebras. We are concerned here, however, with another facet of the subject based on the observation that very general algebras of continuous func tions tend to exhibit certain properties that are strongly reminiscent of analyticity. Although there exist a variety of well-known properties of this kind that could be mentioned, in many ways the most striking is a local maximum modulus principle proved in 1960 by Hugo Rossi [RIl]. This result, one of the deepest and most elegant in the theory of function algebras, is an essential tool in the theory as we have developed it here. It holds for an arbitrary Banaeh algebra of £unctions defined on the spectrum (maximal ideal space) of the algebra. These are the algebras, along with appropriate generalizations to algebras defined on noncompact spaces, that we call "natural func tion algebras".

Inhalt
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.