Moebius-invariant algebras in balls

Walter Rudin

Annales de l'institut Fourier (1983)

  • Volume: 33, Issue: 2, page 19-41
  • ISSN: 0373-0956

Abstract

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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 ) .

How to cite

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Rudin, Walter. "Moebius-invariant algebras in balls." Annales de l'institut Fourier 33.2 (1983): 19-41. <http://eudml.org/doc/74585>.

@article{Rudin1983,
abstract = {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 $f\circ \Psi \in Y$ whenever $f\in Y$ and $\Psi $ is a biholomorphic map of the open unit ball $B$ of $\{\bf C\}^n$ 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 $C(B)$.},
author = {Rudin, Walter},
journal = {Annales de l'institut Fourier},
keywords = {Moebius-invariant algebras; Frechet algebra},
language = {eng},
number = {2},
pages = {19-41},
publisher = {Association des Annales de l'Institut Fourier},
title = {Moebius-invariant algebras in balls},
url = {http://eudml.org/doc/74585},
volume = {33},
year = {1983},
}

TY - JOUR
AU - Rudin, Walter
TI - Moebius-invariant algebras in balls
JO - Annales de l'institut Fourier
PY - 1983
PB - Association des Annales de l'Institut Fourier
VL - 33
IS - 2
SP - 19
EP - 41
AB - 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 $f\circ \Psi \in Y$ whenever $f\in Y$ and $\Psi $ is a biholomorphic map of the open unit ball $B$ of ${\bf C}^n$ 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 $C(B)$.
LA - eng
KW - Moebius-invariant algebras; Frechet algebra
UR - http://eudml.org/doc/74585
ER -

References

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  2. [2] M. L. AGRANOVSKII, Invariant algebras on noncompact Riemannian symmetric spaces, Soviet Math. Dokl., 13 (1972), 1538-1542. Zbl0279.46026MR47 #2276
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  8. [8] F. JOHN, Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience, 1955. Zbl0067.32101MR17,746d
  9. [9] K. de LEEUW and H. MIRKIL, Rotation-invariant algebras on the n-sphere, Duke Math. J., 30 (1963), 667-672. Zbl0194.16101MR27 #5117
  10. [10] A. NAGEL and W. RUDIN, Moebius-invariant function spaces on balls and spheres, Duke Math. J., 43 (1976), 841-865. Zbl0343.32017MR54 #13135
  11. [11] W. RUDIN, Function Theory in the Unit Ball of Cn, Springer, 1980. Zbl0495.32001MR82i:32002
  12. [12] W. RUDIN, Functional Analysis, Mc Graw-Hill, 1973. Zbl0253.46001MR51 #1315
  13. [13] G. STOLZENBERG, Uniform approximation on smooth curves, Acta Math., 115 (1966), 185-198. Zbl0143.30005MR33 #307
  14. [14] E. L. STOUT, The Theory of Uniform Algebras, Bogden and Quigley, 1971. Zbl0286.46049MR54 #11066

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