Illinois Journal of Mathematics

A characterization of $C(K)$ among function algebras on a Riemann surface

Lynette J. Boos

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Abstract

For a compact subset $K$ of a Riemann surface, necessary and sufficient conditions are given for a function algebra containing $A(K)$ to be all of $C(K)$. Using these results, several conditions are given on a complex-valued function $f$ so that the algebra generated by $A(K)$ and $f$ is all of $C(K)$. In particular, the results are applied to a harmonic function $f$ to give sufficient conditions for the algebra generated by $A(K)$ and $f$ to be all of $C(K)$. Also, sufficient conditions are given for the algebra $A(K)$ to be a maximal subalgebra of $C(K)$.

Article information

Source
Illinois J. Math., Volume 52, Number 3 (2008), 867-885.

Dates
First available in Project Euclid: 1 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1254403719

Digital Object Identifier
doi:10.1215/ijm/1254403719

Mathematical Reviews number (MathSciNet)
MR2546012

Zentralblatt MATH identifier
1192.46048

Subjects
Primary: 46J10: Banach algebras of continuous functions, function algebras [See also 46E25] 32A38: Algebras of holomorphic functions [See also 30H05, 46J10, 46J15]
Secondary: 30F15: Harmonic functions on Riemann surfaces

Citation

Boos, Lynette J. A characterization of $C(K)$ among function algebras on a Riemann surface. Illinois J. Math. 52 (2008), no. 3, 867--885. doi:10.1215/ijm/1254403719. https://projecteuclid.org/euclid.ijm/1254403719


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References

  • H. Alexander and J. Wermer, Several complex variables and Banach algebras, third ed. Graduate Texts in Mathematics, vol. 35, Springer, New York, 1998.
  • S. Axler and A. Shields, Algebras generated by analytic and harmonic functions, Indiana Univ. Math. J. 36 (1987), 631–638.
  • H. Behnke and K. Stein, Entwicklung analytischer Funktionen auf Riemannschen Flächen, Math. Ann. 120 (1949), 430–461.
  • A. Browder, Introduction to function algebras, W.A. Benjamin, New York, 1969.
  • A. Boivin, T-invariant algebras on Riemann surfaces, Mathematika 34 (1987), 160–171.
  • E. M. ${\rm\check C}$irka, Approximation by holomorphic functions on smooth manifolds in $\mathbb{C}^n$, Math. Sb. 78 (1969), 101–123; English translation, Math. USSR-Sb. 7 (1969), 95–114.
  • M. Freeman, Some conditions for uniform approximation on a manifold, Function algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965), Scott, Foresman and Company, Chicago, IL, 1966, pp. 42–60.
  • T. W. Gamelin, Uniform algebras, 2nd ed., Chelsea Publishing Company, New York, 1984.
  • T. W. Gamelin, Uniform algebras and Jensen measures, London Math. Soc., Lecture Notes Series, vol. 32, Cambridge Univ. Press, Cambridge, 1978.
  • T. W. Gamelin and H. Rossi, Jensen measures and algebras of analytic functions, Function algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965), Scott, Foresman and Co., Chicago, IL, 1966, pp. 15–35.
  • V. Guillemin and A. Pollack, Differential topology, Prentice-Hall Inc., Englewood Cliffs, NJ, 1974.
  • R. C. Gunning and R. Narasimhan, Immersion of open Riemann surfaces, Math. Ann. 174 (1967), 103–108.
  • P. M. Gauthier, Meromorphic uniform approximation on closed subsets of open Riemann surfaces, Approximation theory and functional analysis (Proc. Internat. Sympos. Approximation Theory, Univ. Estadual de Campinas, Campinas, 1977), North-Holland Math. Stud., vol. 35, North-Holland Publishing Co., Amsterdam, 1979, pp. 139–158.
  • A. J. Izzo, Uniform approximation by holomorphic and harmonic functions, J. London Math. Soc. (2) 47 (1993), 129–141.
  • A. J. Izzo, A characterization of $C(K)$ among the uniform algebras containing $A(K)$, Indiana Univ. Math. J. 46 (1997), 771–788.
  • A. J. Izzo, Algebras containing bounded holomorphic functions, Indiana Univ. Math. J. 52 (2003), 1305–1342.
  • B. Jiang, Uniform approximation on Riemann surfaces by holomorphic and harmonic functions, Illinois J. Math. 47 (2003), 1099–1113.
  • L. K. Kodama, Boundary measure of analytic differentials and uniform approximation on a Riemann surface, Pacific J. Math. 15 (1965), 1261–1277.
  • R. Narasimhan, Analysis on real and complex manifolds, Advanced Studies in Pure Mathematics, vol. 1, North-Holland Publishing Co., Amsterdam, 1973.
  • A. Sakai, Localization theorem for holomorphic approximation on open Riemann surfaces, J. Math. Soc. Japan 24 (1972), 189–197.
  • S. Scheinberg, Uniform approximation by functions analytic on a Riemann surface, Ann. of Math. (2) 108 (1978), 257–298.
  • E. L. Stout, The theory of uniform algebras, Bogden and Quigley, Inc., Tarrytown-on-Hudson, NY, 1971.
  • J. Wermer, Polynomially convex disks, Math. Ann. 158 (1965), 6–10.
  • J. Wermer, On algebras of functions, Proc. Amer. Math. Soc. 4 (1953), 866–869.