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Transactions of the American Mathematical Society

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Analytic equivalence among simply connected domains in $ C(X)$


Author: Hugh E. Warren
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
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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 [Enhancements On Off] (What's this?)

  • [1] L. V. Ahlfors and L. Sario, Riemann surfaces, Princeton Math. Series, no. 26, Princeton Univ. Press, Princeton N. J., 1960, p. 180. MR 22 #5729. MR 0114911 (22:5729)
  • [2] B. W. Glickfeld, On the inverse function theorem in commutative Banach algebras, Illinois J. Math. 15 (1971), 212-221. MR 42 #8287. MR 0273408 (42:8287)
  • [3] G. M. Goluzin, Geometric theory of functions of a complex variable, GITTL, Moscow, 1952; English transl., Transl. Math Monographs, vol. 26, Amer. Math. Soc., Providence, R. I., 1969, p. 54. MR 15, 112; 40 #308. MR 0247039 (40:308)
  • [4] M. Heins, Complex function theory, Pure and Appl. Math., vol. 28, Academic Press, New York, 1968, p. 343. MR 39 #413. MR 0239054 (39:413)
  • [5] E. Hille, Analytic function theory. Vol II, Introduction to Higher Math., Ginn, Boston, Mass., 1962, p. 236. MR 34 #1490. MR 0201608 (34:1490)
  • [6] S.-T. Hu, Elements of general topology, Holden-Day, San Francisco, Calif., 1964, p. 71. MR 31 #1643. MR 0177380 (31:1643)
  • [7] E. R. Lorch, The theory of analytic functions in normed abelian vector rings, Trans. Amer. Math. Soc. 54 (1943), 414-425. MR 5, 100. MR 0009090 (5:100a)
  • [8] S. Saks and A. Zygmund, Analytic functions, 2nd ed., Monografie Mat., Tom 10, PWN, Warsaw, 1938; English transl., Monografie, Mat., Tom 28, PWN, Warsaw, 1965, p. 231. MR 31 #4889. MR 0055432 (14:1073a)
  • [9] H. E. Warren, A Riemann mapping theorem for $ C(X)$, Proc. Amer. Math. Soc. 28 (1971), 147-154. MR 43 #5300. MR 0279578 (43:5300)
  • [10] -, Sets in $ C(X)$ analytically equivalent to the open ball, Duke Math. J. 39 (1972), 711-717. MR 0352981 (50:5467)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0358349-6
Keywords: Lorch analytic function, analytic equivalence, Riemann surface
Article copyright: © Copyright 1974 American Mathematical Society

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