Analytic equivalence among simply connected domains in $C(X)$
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- by Hugh E. Warren
- Trans. Amer. Math. Soc. 196 (1974), 265-288
- DOI: https://doi.org/10.1090/S0002-9947-1974-0358349-6
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Abstract:
This work considers analytic equivalence within the analytic function theory for commutative Banach algebras which was introduced by E. R. Lorch. Necessary conditions of a geometric nature are given for simply connected domains in $C(X)$. These show that there are a great many equivalence classes. In some important cases, as when one domain is the unit ball, the given conditions are also sufficient. The main technique is the association of a simply connected domain in $C(X)$ with a family of Riemann surfaces over the plane.References
- Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911, DOI 10.1515/9781400874538
- Barnett W. Glickfeld, On the inverse function theorem in commutative Banach algebras, Illinois J. Math. 15 (1971), 212–221. MR 273408
- G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039, DOI 10.1090/mmono/026
- Maurice Heins, Complex function theory, Pure and Applied Mathematics, Vol. 28, Academic Press, New York-London, 1968. MR 0239054
- Einar Hille, Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Company, Boston, Mass.-New York-Toronto, 1962. MR 0201608
- Sze-tsen Hu, Elements of general topology, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964. MR 0177380
- Edgar R. Lorch, The theory of analytic functions in normed Abelian vector rings, Trans. Amer. Math. Soc. 54 (1943), 414–425. MR 9090, DOI 10.1090/S0002-9947-1943-0009090-0
- Stanisław Saks and Antoni Zygmund, Analytic functions, Monografie Matematyczne, Tom XXVIII, Polskie Towarzystwo Matematyczne, Warszawa-Wroclaw, 1952. Translated by E. J. Scott. MR 0055432
- Hugh E. Warren, A Riemann mapping theorem for $C(X)$, Proc. Amer. Math. Soc. 28 (1971), 147–154. MR 279578, DOI 10.1090/S0002-9939-1971-0279578-9
- H. E. Warren, Sets in $C(X)$ analytically equivalent to the open ball, Duke Math. J. 39 (1972), 711–717. MR 352981, DOI 10.1215/S0012-7094-72-03976-2
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 196 (1974), 265-288
- MSC: Primary 46G20; Secondary 30A98, 46J10
- DOI: https://doi.org/10.1090/S0002-9947-1974-0358349-6
- MathSciNet review: 0358349