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.
-
[1]
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.
@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},
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MRNUMBER = {1815422},
DOI = {10.1112/blms/33.2.187},
ZBLNUMBER = {1041.32021},
} -
[2]
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.
@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]
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.
@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]
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.
@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]
B. Berndtsson, "Integral kernels and approximation on totally real submanifolds of ${\bf C}^{n}$," Math. Ann., vol. 243, iss. 2, pp. 125-129, 1979.
@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.
@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.
@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]
M. Freeman, "Uniform approximation on a real-analytic manifold," Trans. Amer. Math. Soc., vol. 143, pp. 545-553, 1969.
@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.
@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.
@book{10,
author={Gamelin, T. W.},
TITLE={Uniform Algebras},
EDITION={2nd},
PUBLISHER={Chelsea},
ADDRESS={New York},
YEAR={1984},
MRNUMBER = {0410387},
ZBLNUMBER = {0213.40401},
} -
[11]
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.
@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.
@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]
A. J. Izzo, "Algebras containing bounded holomorphic functions," Indiana Univ. Math. J., vol. 52, iss. 5, pp. 1305-1342, 2003.
@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]
A. J. Izzo, "Uniform algebras on the sphere invariant under group actions," Math. Ann., vol. 344, iss. 4, pp. 989-995, 2009.
@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]
A. J. Izzo, "Uniform algebras invariant under transitive group actions," Indiana Univ. Math. J., vol. 59, iss. 2, pp. 417-426, 2010.
@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.
@book{16,
author={Munkres, J. R.},
TITLE={Topology},
EDITION={2nd},
PUBLISHER={Prentice-Hall},
ADDRESS={Upper Saddle River, NJ},
YEAR={2000},
ZBLNUMBER = {0951.54001},
} -
[17]
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.
@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]
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.
@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.
@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.
@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]
B. M. Weinstock, "Inhomogeneous Cauchy-Riemann systems with smooth dependence on parameters," Duke Math. J., vol. 40, pp. 307-312, 1973.
@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]
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.
@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},
} -
@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},
} -
@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.
@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},
}