Spaces of type H +C
Annales de l'Institut Fourier, Tome 25 (1975) no. 1, pp. 99-125.

On démontre un théorème facile concernant une condition suffisante pour que la somme de deux sous-espaces fermés d’un espace de Banach soit fermée. Ce théorème conduit à plusieurs résultats du type du théorème de Sarason, qui dit que H +C est une sous-algèbre fermée de L . Dans ces résultats, le cercle est remplacé par d’autres groupes, et au lieu du disque unité on considère les polydisques et boules dans les espaces de plusieurs variables complexes. Les sommes des idéaux fermés dans une algèbre de Banach sont aussi étudiés.

A simple theorem is proved which states a sufficient condition for the sum ot two closed subspaces of a Banach space to be closed. This leads to several analogues of Sarason’s theorem which states that H +C is a closed subalgebra of L . In these analogues, the unit circle is replaces by other groups, and the unit disc is replaced by polydiscs or by balls in spaces of several complex variables. Sums of closed ideals in Banach algebras are also studied.

@article{AIF_1975__25_1_99_0,
     author = {Rudin, Walter},
     title = {Spaces of type $H^\infty +C$},
     journal = {Annales de l'Institut Fourier},
     pages = {99--125},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {25},
     number = {1},
     year = {1975},
     doi = {10.5802/aif.545},
     mrnumber = {51 #13692},
     zbl = {0295.46080},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.545/}
}
TY  - JOUR
AU  - Rudin, Walter
TI  - Spaces of type $H^\infty +C$
JO  - Annales de l'Institut Fourier
PY  - 1975
SP  - 99
EP  - 125
VL  - 25
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.545/
DO  - 10.5802/aif.545
LA  - en
ID  - AIF_1975__25_1_99_0
ER  - 
%0 Journal Article
%A Rudin, Walter
%T Spaces of type $H^\infty +C$
%J Annales de l'Institut Fourier
%D 1975
%P 99-125
%V 25
%N 1
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.545/
%R 10.5802/aif.545
%G en
%F AIF_1975__25_1_99_0
Rudin, Walter. Spaces of type $H^\infty +C$. Annales de l'Institut Fourier, Tome 25 (1975) no. 1, pp. 99-125. doi : 10.5802/aif.545. http://www.numdam.org/articles/10.5802/aif.545/

[1] L. Bungart, Boundary kernel functions for domains on complex manifolds, Pacific J. Math., 14 (1960), 1151-1164. | MR | Zbl

[2] F. Combes, Sur les faces d'une C*-algèbre, Bull. Sci. Math., 93 (1969), 37-62. | MR | Zbl

[3] A.M. Davie, T.W. Gamelin, and J. Garnett, Distance estimates and pointwise bounded density, Trans. Amer. Math. Soc., 175 (1973), 37-68. | MR | Zbl

[4] A. Devinatz, An extension of a limit theorem of G. Szegö, J. Math. Anal. Appl., 14 (1966), 499-510. | MR | Zbl

[5] J. Dixmier, Les C*-algèbres et leurs Représentations, Gauthier-Villars, Paris, 1969. | MR | Zbl

[6] F. Forelli, Measures whose Poisson integrals are pluriharmonic, Illinois J. Math., 18 (1974), 373-388. | MR | Zbl

[7] H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables, Acta Math., 99 (1958), 165-202. | MR | Zbl

[8] H. Helson and D. Sarason, Past and future, Math. Scand., 21 (1967), 5-16. | MR | Zbl

[9] G.M. Henkin, Integral representations of functions holomorphic in strictly pseudoconvex domains and some applications, Math. USSR Sbornik, 7 (1969), 597-616. (Mat. Sbornik 78 (1969)). | Zbl

[10] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis, Springer Verlag, Berlin ; Vol. 1, 1963 ; Vol. 2, 1970.

[11] A. Koranyi, Harmonic functions on hermitian hyperbolic space, Trans. Amer. Math. Soc., 135 (1969), 507-516. | MR | Zbl

[12] A. Koranyi and S. Vagi, Singular integrals in homogeneous spaces and some problems of classical analysis, Ann. Scuola Normale Superiore Pisa, 25 (1971), 575-648. | Numdam | MR | Zbl

[13] J.T. Marti, Introduction to the Theory of Bases, Springer Verlag, 1969. | MR | Zbl

[14] C.E. Rickart, General Theory of Banach Algebras, Van Nostrand, 1960. | MR | Zbl

[15] W. Rudin, The closed ideals in an algebra of analytic functions, Can. J. Math., 9 (1957), 426-434. | MR | Zbl

[16] W. Rudin, Fourier Analysis on Groups, Interscience, 1962. | MR | Zbl

[17] W. Rudin, Function Theory in Polydiscs, Benjamin, 1969. | MR | Zbl

[18] D. Sarason, Generalized interpolation in H∞, Trans. Amer. Math. Soc., 127 (1967), 179-203. | MR | Zbl

[19] D. Sarason, Algebras of functions on the unit circle, Bull. Amer, Math. Soc., 79 (1973), 286-299. | MR | Zbl

[20] E.M. Stein, Boundary Behavior of Holomorphic Functions of Several Complex Variables, Princeton University Press, 1972. | MR | Zbl

[21] E.L. Stout, On the multiplicative Cousin problem with bounded data, Ann. Scuola Normale Superiore Pisa, 27 (1973), 1-17. | Numdam | MR | Zbl

[22] J. Wichmann, Bounded approximate units and bounded approximate identities, Proc. Amer. Math. Soc., 41 (1973), 547-550. | MR | Zbl

[23] L. Zalcman, Bounded analytic functions on domains of infinite connectivity, Trans. Amer. Math. Soc., 144 (1969), 241-269. | MR | Zbl

[24] A. Zygmund, Sur un théorème de M. Fekete, Bull. Acad. Polonaise, (1927), 343-347. | JFM

[25] A. Zygmund, Trigonometric Series, 2nd Ed., Cambridge University Press, 1959. | Zbl

Cité par Sources :