Uniform approximation on manifolds

Alexander J. Izzo

doi:http://doi.org/10.4007/annals.2011.174.1.2

Abstract

It is shown that if $A$ is a uniform algebra generated by a family $\Phi$ of complex-valued $C^1$ functions on a compact $C^1$ manifold-with-boundary $M$, the maximal ideal space of $A$ is $M$, and $E$ is the set of points where the differentials of the functions in $\Phi$ fail to span the complexified cotangent space to $M$, then $A$ contains every continuous function on $M$ that vanishes on $E$. This answers a 45-year-old question of Michael Freeman who proved the special case in which the manifold $M$ is two-dimensional. More general forms of the theorem are also established. The results presented strengthen results due to several mathematicians.

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[1] $Go to document$ J. T. Anderson and A. J. Izzo, "A peak point theorem for uniform algebras generated by smooth functions on two-manifolds," Bull. London Math. Soc., vol. 33, iss. 2, pp. 187-195, 2001.

Show bibtex

@article {1, MRKEY = {1815422}, AUTHOR = {Anderson, John T. and Izzo, Alexander J.}, TITLE = {A peak point theorem for uniform algebras generated by smooth functions on two-manifolds}, JOURNAL = {Bull. London Math. Soc.}, FJOURNAL = {The Bulletin of the London Mathematical Society}, VOLUME = {33}, YEAR = {2001}, NUMBER = {2}, PAGES = {187--195}, ISSN = {0024-6093}, CODEN = {LMSBBT}, MRCLASS = {32T40 (32E30 46J10)}, MRNUMBER = {1815422}, DOI = {10.1112/blms/33.2.187}, ZBLNUMBER = {1041.32021}, }
[2] $Go to document$ J. T. Anderson and A. J. Izzo, "Peak point theorems for uniform algebras on smooth manifolds," Math. Z., vol. 261, iss. 1, pp. 65-71, 2009.

Show bibtex

@article {2, MRKEY = {2452637}, AUTHOR = {Anderson, John T. and Izzo, Alexander J.}, TITLE = {Peak point theorems for uniform algebras on smooth manifolds}, JOURNAL = {Math. Z.}, FJOURNAL = {Mathematische Zeitschrift}, VOLUME = {261}, YEAR = {2009}, NUMBER = {1}, PAGES = {65--71}, ISSN = {0025-5874}, CODEN = {MAZEAX}, MRCLASS = {46J10 (32A65 46J15)}, MRNUMBER = {2452637}, MRREVIEWER = {Keiji Izuchi}, DOI = {10.1007/s00209-008-0313-x}, ZBLNUMBER = {1166.46030}, }
[3] $Go to document$ J. T. Anderson, A. J. Izzo, and J. Wermer, "Polynomial approximation on three-dimensional real-analytic submanifolds of ${\bf C}^n$," Proc. Amer. Math. Soc., vol. 129, iss. 8, pp. 2395-2402, 2001.

Show bibtex

@article {3, MRKEY = {1823924}, AUTHOR = {Anderson, John T. and Izzo, Alexander J. and Wermer, John}, TITLE = {Polynomial approximation on three-dimensional real-analytic submanifolds of {${\bf C}\sp n$}}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {129}, YEAR = {2001}, NUMBER = {8}, PAGES = {2395--2402}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {32E30 (32E20 46J10)}, MRNUMBER = {1823924}, MRREVIEWER = {P. J. de Paepe}, DOI = {10.1090/S0002-9939-01-05911-1}, ZBLNUMBER = {0976.32008}, }
[4] $Go to document$ J. T. Anderson, A. J. Izzo, and J. Wermer, "Polynomial approximation on real-analytic varieties in $\Bbb C^n$," Proc. Amer. Math. Soc., vol. 132, iss. 5, pp. 1495-1500, 2004.

Show bibtex

@article {4, MRKEY = {2053357}, AUTHOR = {Anderson, John T. and Izzo, Alexander J. and Wermer, John}, TITLE = {Polynomial approximation on real-analytic varieties in {$\bold C\sp n$}}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {132}, YEAR = {2004}, NUMBER = {5}, PAGES = {1495--1500}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {32E30}, MRNUMBER = {2053357}, MRREVIEWER = {Joaqu{\'ı}n Ma. Ortega Aramburu}, DOI = {10.1090/S0002-9939-03-07263-0}, ZBLNUMBER = {1058.32006}, }
[5] $Go to document$ B. Berndtsson, "Integral kernels and approximation on totally real submanifolds of ${\bf C}^{n}$," Math. Ann., vol. 243, iss. 2, pp. 125-129, 1979.

Show bibtex

@article {5, MRKEY = {0543722}, AUTHOR = {Berndtsson, Bo}, TITLE = {Integral kernels and approximation on totally real submanifolds of {${\bf C}\sp{n}$}}, JOURNAL = {Math. Ann.}, FJOURNAL = {Mathematische Annalen}, VOLUME = {243}, YEAR = {1979}, NUMBER = {2}, PAGES = {125--129}, ISSN = {0025-5831}, CODEN = {MAANA3}, MRCLASS = {32E30}, MRNUMBER = {0543722}, MRREVIEWER = {Anne-Marie Chollet}, DOI = {10.1007/BF01420419}, ZBLNUMBER = {0394.41012}, }
[6] E. M. vCirka, "Approximation by holomorphic functions on smooth manifolds in ${\bf C}^{n}$," Mat. Sb., vol. 78 (120), pp. 101-123, 1969.

Show bibtex

@article {6, MRKEY = {0239121}, AUTHOR = {{\v{C}}irka, E. M.}, TITLE = {Approximation by holomorphic functions on smooth manifolds in {${\bf C}\sp{n}$}}, JOURNAL = {Mat. Sb.}, VOLUME = {78 (120)}, YEAR = {1969}, NOTE={in {R}ussian; translated in {\it Math.} USSR {\it Sb.} {\bf 7} (1969), 95--114}, PAGES = {101--123}, MRCLASS = {32.70}, MRNUMBER = {0239121}, MRREVIEWER = {R. O. Wells, Jr.}, ZBLNUMBER = {0188.39101}, }
[7] M. Freeman, "Some conditions for uniform approximation on a manifold," in Function Algebras, Chicago, Ill., 1966, pp. 42-60.

Show bibtex

@inproceedings {7, MRKEY = {0193538}, AUTHOR = {Freeman, Michael}, TITLE = {Some conditions for uniform approximation on a manifold}, BOOKTITLE = {Function {A}lgebras}, VENUE={{P}roc. {I}nternat. {S}ympos. on {F}unction {A}lgebras, {T}ulane {U}niv., 1965}, PAGES = {42--60}, PUBLISHER = {Scott-Foresman}, ADDRESS = {Chicago, Ill.}, YEAR = {1966}, MRCLASS = {46.55 (41.00)}, MRNUMBER = {0193538}, MRREVIEWER = {R. E. Edwards}, ZBLNUMBER = {0144.37502}, }
[8] $Go to document$ M. Freeman, "Uniform approximation on a real-analytic manifold," Trans. Amer. Math. Soc., vol. 143, pp. 545-553, 1969.

Show bibtex

@article {8, MRKEY = {0248525}, AUTHOR = {Freeman, Michael}, TITLE = {Uniform approximation on a real-analytic manifold}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {143}, YEAR = {1969}, PAGES = {545--553}, ISSN = {0002-9947}, MRCLASS = {46.55 (32.00)}, MRNUMBER = {0248525}, MRREVIEWER = {W. B. Jones}, DOI = {10.2307/1995263}, ZBLNUMBER = {0188.45101}, }
[9] J. E. Fornaess, "Uniform approximation on manifolds," Math. Scand., vol. 31, pp. 166-170, 1972.

Show bibtex

@article {9, MRKEY = {0344895}, AUTHOR = {Fornaess, John Erik}, TITLE = {Uniform approximation on manifolds}, JOURNAL = {Math. Scand.}, FJOURNAL = {Mathematica Scandinavica}, VOLUME = {31}, YEAR = {1972}, PAGES = {166--170}, ISSN = {0025-5521}, MRCLASS = {46J10 (32E30)}, MRNUMBER = {0344895}, MRREVIEWER = {J. Garnett}, ZBLNUMBER = {0249.32011}, }
[10] T. W. Gamelin, Uniform Algebras, 2nd ed., New York: Chelsea, 1984.

Show bibtex

@book{10, author={Gamelin, T. W.}, TITLE={Uniform Algebras}, EDITION={2nd}, PUBLISHER={Chelsea}, ADDRESS={New York}, YEAR={1984}, MRNUMBER = {0410387}, ZBLNUMBER = {0213.40401}, }
[11] $Go to document$ R. F. Harvey and R. O. Wells Jr., "Holomorphic approximation and hyperfunction theory on a $C^{1}$ totally real submanifold of a complex manifold," Math. Ann., vol. 197, pp. 287-318, 1972.

Show bibtex

@article {11, MRKEY = {0310278}, AUTHOR = {Harvey, F. Reese and Wells, Jr., R. O.}, TITLE = {Holomorphic approximation and hyperfunction theory on a {$C\sp{1}$} totally real submanifold of a complex manifold}, JOURNAL = {Math. Ann.}, FJOURNAL = {Mathematische Annalen}, VOLUME = {197}, YEAR = {1972}, PAGES = {287--318}, ISSN = {0025-5831}, MRCLASS = {32E30}, MRNUMBER = {0310278}, MRREVIEWER = {A. Hirschowitz}, DOI = {10.1007/BF01428202}, ZBLNUMBER = {0246.32019}, }
[12] L. Hörmander and J. Wermer, "Uniform approximation on compact sets in $\mathbf{C}^{n}$," Math. Scand., vol. 23, pp. 5-21 (1969), 1968.

Show bibtex

@article {12, MRKEY = {0254275}, AUTHOR = {H{ö}rmander, L. and Wermer, J.}, TITLE = {Uniform approximation on compact sets in {$\mathbf{C}\sp{n}$}}, JOURNAL = {Math. Scand.}, FJOURNAL = {Mathematica Scandinavica}, VOLUME = {23}, YEAR = {1968}, PAGES = {5--21 (1969)}, ISSN = {0025-5521}, MRCLASS = {32.70 (35.00)}, MRNUMBER = {0254275}, MRREVIEWER = {F. T. Birtel}, ZBLNUMBER = {0181.36201}, }
[13] $Go to document$ A. J. Izzo, "Algebras containing bounded holomorphic functions," Indiana Univ. Math. J., vol. 52, iss. 5, pp. 1305-1342, 2003.

Show bibtex

@article {13, MRKEY = {2010729}, AUTHOR = {Izzo, Alexander J.}, TITLE = {Algebras containing bounded holomorphic functions}, JOURNAL = {Indiana Univ. Math. J.}, FJOURNAL = {Indiana University Mathematics Journal}, VOLUME = {52}, YEAR = {2003}, NUMBER = {5}, PAGES = {1305--1342}, ISSN = {0022-2518}, CODEN = {IUMJAB}, MRCLASS = {46J15 (30F15 30H05)}, MRNUMBER = {2010729}, MRREVIEWER = {Raymond Mortini}, DOI = {10.1512/iumj.2003.52.2315}, ZBLNUMBER = {1093.46025}, }
[14] $Go to document$ A. J. Izzo, "Uniform algebras on the sphere invariant under group actions," Math. Ann., vol. 344, iss. 4, pp. 989-995, 2009.

Show bibtex

@article {14, MRKEY = {2507636}, AUTHOR = {Izzo, Alexander J.}, TITLE = {Uniform algebras on the sphere invariant under group actions}, JOURNAL = {Math. Ann.}, FJOURNAL = {Mathematische Annalen}, VOLUME = {344}, YEAR = {2009}, NUMBER = {4}, PAGES = {989--995}, ISSN = {0025-5831}, CODEN = {MAANA}, MRCLASS = {46J10 (46J15)}, MRNUMBER = {2507636}, MRREVIEWER = {T. Tonev}, DOI = {10.1007/s00208-009-0349-1}, ZBLNUMBER = {1184.46049}, }
[15] $Go to document$ A. J. Izzo, "Uniform algebras invariant under transitive group actions," Indiana Univ. Math. J., vol. 59, iss. 2, pp. 417-426, 2010.

Show bibtex

@article {15, MRKEY = {2648073}, AUTHOR = {Izzo, Alexander J.}, TITLE = {Uniform algebras invariant under transitive group actions}, JOURNAL = {Indiana Univ. Math. J.}, FJOURNAL = {Indiana University Mathematics Journal}, VOLUME = {59}, YEAR = {2010}, NUMBER = {2}, PAGES = {417--426}, ISSN = {0022-2518}, CODEN = {IUMJAB}, MRCLASS = {46J10}, MRNUMBER = {2648073}, ZBLNUMBER = {05808400}, DOI = {10.1512/iumj.2010.59.4032}, }
[16] J. R. Munkres, Topology, 2nd ed., Upper Saddle River, NJ: Prentice-Hall, 2000.

Show bibtex

@book{16, author={Munkres, J. R.}, TITLE={Topology}, EDITION={2nd}, PUBLISHER={Prentice-Hall}, ADDRESS={Upper Saddle River, NJ}, YEAR={2000}, ZBLNUMBER = {0951.54001}, }
[17] $Go to document$ R. Nirenberg and R. O. Wells Jr., "Holomorphic approximation on real submanifolds of a complex manifold," Bull. Amer. Math. Soc., vol. 73, pp. 378-381, 1967.

Show bibtex

@article {17, MRKEY = {0209850}, AUTHOR = {Nirenberg, Ricardo and Wells, Jr., R. O.}, TITLE = {Holomorphic approximation on real submanifolds of a complex manifold}, JOURNAL = {Bull. Amer. Math. Soc.}, FJOURNAL = {Bulletin of the American Mathematical Society}, VOLUME = {73}, YEAR = {1967}, PAGES = {378--381}, ISSN = {0002-9904}, MRCLASS = {46.65 (32.00)}, MRNUMBER = {0209850}, MRREVIEWER = {J. Kajiwara}, DOI = {10.1090/S0002-9904-1967-11760-9}, ZBLNUMBER = {0162.10402}, }
[18] $Go to document$ R. Nirenberg and R. O. Wells Jr., "Approximation theorems on differentiable submanifolds of a complex manifold," Trans. Amer. Math. Soc., vol. 142, pp. 15-35, 1969.

Show bibtex

@article {18, MRKEY = {0245834}, AUTHOR = {Nirenberg, Ricardo and Wells, Jr., R. O.}, TITLE = {Approximation theorems on differentiable submanifolds of a complex manifold}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {142}, YEAR = {1969}, PAGES = {15--35}, ISSN = {0002-9947}, MRCLASS = {32.40}, MRNUMBER = {0245834}, MRREVIEWER = {M. Herv{é}}, DOI = {10.2307/1995342}, ZBLNUMBER = {0188.39103}, }
[19] A. G. O’Farrell, K. J. Preskenis, and D. Walsh, "Holomorphic approximation in Lipschitz norms," in Proceedings of the Conference on Banach Algebras and Several Complex Variables, Providence, RI, 1984, pp. 187-194.

Show bibtex

@inproceedings {19, MRKEY = {0769507}, AUTHOR = {O'Farrell, A. G. and Preskenis, K. J. and Walsh, D.}, TITLE = {Holomorphic approximation in {L}ipschitz norms}, BOOKTITLE = {Proceedings of the Conference on {B}anach Algebras and Several Complex Variables}, VENUE={{N}ew {H}aven, {C}onn., 1983}, SERIES = {Contemp. Math.}, VOLUME = {32}, PAGES = {187--194}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {1984}, MRCLASS = {32E30 (32E05)}, MRNUMBER = {0769507}, MRREVIEWER = {Wies{\l}aw Ple{\'s}niak}, ZBLNUMBER = {0553.32015}, }
[20] M. R. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, New York: Springer-Verlag, 1986, vol. 108.

Show bibtex

@book {20, MRKEY = {0847923}, AUTHOR = {Range, R. Michael}, TITLE = {Holomorphic Functions and Integral Representations in Several Complex Variables}, SERIES = {Graduate Texts in Math.}, VOLUME = {108}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1986}, PAGES = {xx+386}, ISBN = {0-387-96259-X}, MRCLASS = {32-01}, MRNUMBER = {0847923}, MRREVIEWER = {Harold P. Boas}, ZBLNUMBER = {0591.32002}, }
[21] $Go to document$ B. M. Weinstock, "Inhomogeneous Cauchy-Riemann systems with smooth dependence on parameters," Duke Math. J., vol. 40, pp. 307-312, 1973.

Show bibtex

@article {21, MRKEY = {0313646}, AUTHOR = {Weinstock, Barnet M.}, TITLE = {Inhomogeneous {C}auchy-{R}iemann systems with smooth dependence on parameters}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {40}, YEAR = {1973}, PAGES = {307--312}, ISSN = {0012-7094}, MRCLASS = {35N15 (32F15)}, MRNUMBER = {0313646}, MRREVIEWER = {J. Kajiwara}, DOI = {10.1215/S0012-7094-73-04026-X}, ZBLNUMBER = {0294.35058}, }
[22] $Go to document$ B. M. Weinstock, "Uniform approximation on the graph of a smooth map in ${\bf C}^{n}$," Canad. J. Math., vol. 32, iss. 6, pp. 1390-1396, 1980.

Show bibtex

@article {22, MRKEY = {0604694}, AUTHOR = {Weinstock, Barnet M.}, TITLE = {Uniform approximation on the graph of a smooth map in {${\bf C}\sp{n}$}}, JOURNAL = {Canad. J. Math.}, FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques}, VOLUME = {32}, YEAR = {1980}, NUMBER = {6}, PAGES = {1390--1396}, ISSN = {0008-414X}, CODEN = {CJMAAB}, MRCLASS = {32E30 (41A10)}, MRNUMBER = {0604694}, MRREVIEWER = {Eric Amar}, DOI = {10.4153/CJM-1980-109-4}, ZBLNUMBER = {0473.32011}, }
[23] $Go to document$ J. Wermer, "Approximation on a disk," Math. Ann., vol. 155, pp. 331-333, 1964.

Show bibtex

@article {23, MRKEY = {0165386}, AUTHOR = {Wermer, J.}, TITLE = {Approximation on a disk}, JOURNAL = {Math. Ann.}, FJOURNAL = {Mathematische Annalen}, VOLUME = {155}, YEAR = {1964}, PAGES = {331--333}, ISSN = {0025-5831}, MRCLASS = {46.55}, MRNUMBER = {0165386}, MRREVIEWER = {E. A. Bishop}, DOI = {10.1007/BF01354865}, ZBLNUMBER = {0122.06803}, }
[24] $Go to document$ J. Wermer, "Polynomially convex disks," Math. Ann., vol. 158, pp. 6-10, 1965.

Show bibtex

@article {24, MRKEY = {0174968}, AUTHOR = {Wermer, J.}, TITLE = {Polynomially convex disks}, JOURNAL = {Math. Ann.}, FJOURNAL = {Mathematische Annalen}, VOLUME = {158}, YEAR = {1965}, PAGES = {6--10}, ISSN = {0025-5831}, MRCLASS = {46.55 (32.70)}, MRNUMBER = {0174968}, MRREVIEWER = {H. Rossi}, DOI = {10.1007/BF01370392}, ZBLNUMBER = {0124.06404}, }
[25] J. Wermer, Banach Algebras and Several Complex Variables, 2nd ed., New York: Springer-Verlag, 1976, vol. 35.

Show bibtex

@book {25, MRKEY = {0394218}, AUTHOR = {Wermer, J.}, TITLE = {Banach Algebras and Several Complex Variables}, EDITION = {2nd}, SERIES = {Graduate Texts in Math.}, VOLUME={35}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1976}, PAGES = {ix+161}, MRCLASS = {46J15 (32E30)}, MRNUMBER = {0394218}, MRREVIEWER = {Editiors}, ZBLNUMBER = {0336.46055}, }

Uniform approximation on manifolds

Abstract

Authors