Moebius-invariant algebras in balls
Rudin, Walter
Annales de l'Institut Fourier, Tome 33 (1983), p. 19-41 / Harvested from Numdam
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On démontre que dans l’algèbre de Fréchet C(B) il y a exactement trois sous-algèbres Y qui sont fermées, qui contiennent des fonctions non constantes, et qui sont invariantes dans le sens suivant : f∘Ψ∈Y lorsque f∈Y et Ψ est une application biholomorphe de la boule unité ouverte B de C n sur B. Ce sont (i) l’algèbre des fonctions holomorphes dans B, (ii) l’algèbre des fonctions f dont les conjuguées f ¯ sont holomorphes, (iii) C(B).

It is proved that the Fréchet algebra C(B) has exactly three closed subalgebras Y which contain nonconstant functions and which are invariant, in the sense that f∘Ψ∈Y whenever f∈Y and Ψ is a biholomorphic map of the open unit ball B of C n onto B. One of these consists of the holomorphic functions in B, the second consists of those whose complex conjugates are holomorphic, and the third is C(B).

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     author = {Rudin, Walter},
     title = {Moebius-invariant algebras in balls},
     journal = {Annales de l'Institut Fourier},
     volume = {33},
     year = {1983},
     pages = {19-41},
     doi = {10.5802/aif.914},
     mrnumber = {84m:32023},
     zbl = {0487.32012},
     mrnumber = {699485},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1983__33_2_19_0}
}

Rudin, Walter. Moebius-invariant algebras in balls. Annales de l'Institut Fourier, Tome 33 (1983) pp. 19-41. doi : 10.5802/aif.914. http://gdmltest.u-ga.fr/item/AIF_1983__33_2_19_0/

[1] M. L. Agranovskii, Invariant algebras on the boundaries of symmetric domains, Soviet Math. Dokl., 12 (1971), 371-374. | Zbl 0225.32012

[2] M. L. Agranovskii, Invariant algebras on noncompact Riemannian symmetric spaces, Soviet Math. Dokl., 13 (1972), 1538-1542. | MR 47 #2276 | Zbl 0279.46026

[3] M. L. Agranovskii and R. E. Valskii, Maximality of invariant algebras of functions, Sib. Math. J., 12 (1971), 1-7. | MR 44 #3128 | Zbl 0226.46059

[4] H. Alexander, Polynomial approximation and hulls of sets of finite linear measure in Cn, Amer. J. Math., 93 (1971), 65-75. | MR 44 #1841 | Zbl 0221.32011

[5] C. A. Berenstein and L. Zalcman, Pompeiu's problem on spaces of constant curvature, J. d'Anal. Math., 30 (1976), 113-130. | MR 55 #11172 | Zbl 0332.35033

[6] C. A. Berenstein and L. Zalcman, Pompeiu's problem on symmetric spaces, Comment. Math. Helvetici, 55 (1980), 593-621. | MR 83d:43012 | Zbl 0452.43012

[7] R. Courant and D. Hilbert, Methoden der Mathematischen Physik, vol. II, Springer, 1937. | JFM 63.0449.05 | Zbl 0017.39702

[8] F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience, 1955. | MR 17,746d | Zbl 0067.32101

[9] K. De Leeuw and H. Mirkil, Rotation-invariant algebras on the n-sphere, Duke Math. J., 30 (1963), 667-672. | MR 27 #5117 | Zbl 0194.16101

[10] A. Nagel and W. Rudin, Moebius-invariant function spaces on balls and spheres, Duke Math. J., 43 (1976), 841-865. | MR 54 #13135 | Zbl 0343.32017

[11] W. Rudin, Function Theory in the Unit Ball of Cn, Springer, 1980. | MR 82i:32002 | Zbl 0495.32001

[12] W. Rudin, Functional Analysis, Mc Graw-Hill, 1973. | MR 51 #1315 | Zbl 0253.46001

[13] G. Stolzenberg, Uniform approximation on smooth curves, Acta Math., 115 (1966), 185-198. | MR 33 #307 | Zbl 0143.30005

[14] E. L. Stout, The Theory of Uniform Algebras, Bogden and Quigley, 1971. | MR 54 #11066 | Zbl 0286.46049