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A characterization of \(C(K)\) among function algebras on a Riemann surface. (English) Zbl 1192.46048

Let \(K\) be a compact subset of an open Riemann surface. Denote by \(C(K)\) and \(A(K)\) the algebra of all complex-valued continuous functions on \(K\) and its subalgebra consisting of those functions which are holomorphic on the interior of \(K\), respectively.
The author presents necessary and sufficient conditions 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 uniformly closed subalgebra generated by \(A(K)\) and \(f\) is all of \(C(K)\). In particular, the author considers the cases when \(f\) is continuously differentiable or harmonic on the interior of \(K\), or it is harmonic with respect to \(A(K)\). Finally, sufficient conditions are given for the algebra \(A(K)\) to be a maximal subalgebra of \(C(K)\). The presented results generalize several of A. J. Izzo’s former results from compact subsets of the complex plane to compact subsets of a Riemann surface.

MSC:

46J10 Banach algebras of continuous functions, function algebras
32A38 Algebras of holomorphic functions of several complex variables
30F15 Harmonic functions on Riemann surfaces
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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References:

[1] H. Alexander and J. Wermer, Several complex variables and Banach algebras , third ed. Graduate Texts in Mathematics, vol. 35, Springer, New York, 1998. · Zbl 0894.46037
[2] S. Axler and A. Shields, Algebras generated by analytic and harmonic functions , Indiana Univ. Math. J. 36 (1987), 631–638. · Zbl 0616.46048
[3] H. Behnke and K. Stein, Entwicklung analytischer Funktionen auf Riemannschen Flä chen, Math. Ann. 120 (1949), 430–461. · Zbl 0038.23502
[4] A. Browder, Introduction to function algebras , W.A. Benjamin, New York, 1969. · Zbl 0199.46103
[5] A. Boivin, T-invariant algebras on Riemann surfaces , Mathematika 34 (1987), 160–171. · Zbl 0619.30040
[6] E. M. \(\mathrm\check C\)irka, Approximation by holomorphic functions on smooth manifolds in \(\mathbbC^n\) , Math. Sb. 78 (1969), 101–123; English translation, Math. USSR-Sb. 7 (1969), 95–114.
[7] 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. · Zbl 0144.37502
[8] T. W. Gamelin, Uniform algebras , 2nd ed., Chelsea Publishing Company, New York, 1984. · Zbl 0213.40401
[9] T. W. Gamelin, Uniform algebras and Jensen measures , London Math. Soc., Lecture Notes Series, vol. 32, Cambridge Univ. Press, Cambridge, 1978. · Zbl 0418.46042
[10] 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. · Zbl 0144.37403
[11] V. Guillemin and A. Pollack, Differential topology , Prentice-Hall Inc., Englewood Cliffs, NJ, 1974. · Zbl 0361.57001
[12] R. C. Gunning and R. Narasimhan, Immersion of open Riemann surfaces , Math. Ann. 174 (1967), 103–108. · Zbl 0179.11402
[13] 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, · Zbl 0409.30033
[14] A. J. Izzo, Uniform approximation by holomorphic and harmonic functions , J. London Math. Soc. (2) 47 (1993), 129–141. · Zbl 0815.46041
[15] A. J. Izzo, A characterization of \(C(K)\) among the uniform algebras containing \(A(K)\) , Indiana Univ. Math. J. 46 (1997), 771–788. · Zbl 0912.46053
[16] A. J. Izzo, Algebras containing bounded holomorphic functions , Indiana Univ. Math. J. 52 (2003), 1305–1342. · Zbl 1093.46025
[17] B. Jiang, Uniform approximation on Riemann surfaces by holomorphic and harmonic functions , Illinois J. Math. 47 (2003), 1099–1113. · Zbl 1040.30020
[18] L. K. Kodama, Boundary measure of analytic differentials and uniform approximation on a Riemann surface , Pacific J. Math. 15 (1965), 1261–1277. · Zbl 0136.06703
[19] R. Narasimhan, Analysis on real and complex manifolds , Advanced Studies in Pure Mathematics, vol. 1, North-Holland Publishing Co., Amsterdam, 1973. · Zbl 0188.25803
[20] A. Sakai, Localization theorem for holomorphic approximation on open Riemann surfaces , J. Math. Soc. Japan 24 (1972), 189–197. · Zbl 0234.30038
[21] S. Scheinberg, Uniform approximation by functions analytic on a Riemann surface , Ann. of Math. (2) 108 (1978), 257–298. JSTOR: · Zbl 0423.30035
[22] E. L. Stout, The theory of uniform algebras , Bogden and Quigley, Inc., Tarrytown-on-Hudson, NY, 1971. · Zbl 0286.46049
[23] J. Wermer, Polynomially convex disks , Math. Ann. 158 (1965), 6–10. · Zbl 0124.06404
[24] J. Wermer, On algebras of functions , Proc. Amer. Math. Soc. 4 (1953), 866–869. JSTOR: · Zbl 0052.12105
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