Author	Rickart, Charles E. author
Title	Natural Function Algebras [electronic resource] / by Charles E. Rickart
Imprint	New York, NY : Springer New York, 1979
Connect to	http://dx.doi.org/10.1007/978-1-4613-8070-2
Descript	XIV, 240 p. 1 illus. online resource

The term "function algebra" usually refers to a uniformly closed algebra of complex valued continuous functions on a compact Hausdorff space. Such Banach algeยญ 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 reยญ 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 funcยญ 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 ยฃunctions 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 funcยญ tion algebras"

I. The Category of Pairs -- ยง 1. Pairs and systems -- ยง 2. Morphisms and extensions of pairs -- ยง 3. Natural systems -- ยง 4. Products of pairs -- ยง 5. Examples and remarks -- II. Convexity and Naturality -- ยง 6. a-convex hulls. Hull-kernel topology -- ยง 7. a-convexity in a natural pair [?, a] -- ยง 8. Closure operations -- ยง 9. Convexity and extensions -- ยง10. Natural extensions -- ยง11. Examples -- III. The ล ilov Boundary and Local Maximum Principle -- ยง12. Independent points -- ยง13. The ล ilov boundary of a pair -- ยง14. A local maximum principle for natural systems -- ยง15. Applications of the local maximum principle -- IV. Holomorphic Functions -- ยง16. Presheaves of continuous functions -- ยง17. Local extensions, ?-holomo?phic functions -- ยง18. Holomorphic maps -- ยง19. Examples and remarks -- V. Maximum Properties of Holomorphic Functions -- ยง20. A local maximum principle for holomorphic functions -- ยง21. Holomorphic peak sets -- ยง22. a-presheaves -- ยง23. A lemma of Glicksberg -- ยง24. Maximal a-presheaves -- VI. Subharmonic Functions -- ยง25. Plurisubharmonic functions in ?n -- ยง26. Definitions. a-subharmonic functions -- ยง27. Basic properties of a-subharmonic functions -- ยง28. Plurisubharmonicity -- ยง29. Maximum properties -- ยง30. Integral representations -- ยง31. Characterization of a-harmonic functions -- ยง32. Hartogโ{128}{153}s functions -- VII. Varieties -- ยง33. Varieties associated with an a-presheaf -- ยง34. Convexity properties -- ยง35. Generalizations of some results of Glicksberg -- ยง36. Continuous families of hypersurfaces -- ยง37. Remarks -- VIII. Holomorphic and Subharmonic Convexity -- ยง38. Convexity with respect to an a-presheaf -- ยง39. Properties of subharmonic convexity -- ยง40. Naturality properties -- ยง41. Holomorphic implied by subharmonic convexity -- ยง42. Local properties -- ยง43. Remarks and an example -- IX. [?, a]-Domains -- ยง44. Definitions -- ยง45. Distance functions -- ยง46. Holomorphic functions -- ยง47. Relative completeness and naturality -- X. Holomorphic Extensions of [?, a]-Domains -- ยง48. Morphisms and extensions. Domains of holomorphy -- ยง49. Existence of maximal extensions -- ยง50. Properties of maximal domains -- ยง51. Remarks -- XI. Holomorphy Theory for Dual Pairs of Vector Spaces -- ยง52. Generalized polynomials and holomorphic functions in a CLTS -- ยง53. Dual pairs ?E, F? -- ยง54. Holomorphic functions in a dual pair -- ยง55. Arens holomorphic functions -- ยง56. Canonical representation of dual pairs -- ยง57. Derivatives -- ยง58. Naturality -- XII. ?E, F? -Domains of Holomorphy -- ยง59. Holomorphic functions in ?E, F?-domains -- ยง60. Subdomains determined by a subspace of F -- ยง61. Envelopes of holomorphy -- ยง62. Series expansions -- ยง63. The finite dimensional component of a domain of holomorphy -- ยง64. The algebra of holomorphic functions -- ยง65. Holomorphic convexity and naturality -- ยง66. A Cartan-Thullen theorem -- XIII. Dual Pair Theory Applied to [?, a]-Domains -- ยง67. The dual pair extension of [?, a]. A-domains -- ยง68. Germ-valued functions -- ยง69. Topologies for [0]? -- ยง70. Naturality of algebras of germ-valued functions -- XIV. Holomorphic Extensions of ?-Domains -- ยง71. Extension relative to germ-valued functions -- ยง72. Uniform families of extensions -- ยง73. Pseudoextensions -- ยง74. Naturality properties -- Index of Symbols -- General Index