Let be a homogeneous algebra on the circle and the closed subalgebra of of functions having analytic extensions into the unit disk . This paper considers the structure of closed ideals of under suitable restrictions on the synthesis properties of . In particular, completely characterized are the closed ideals in whose zero sets meet the circle in a countable set of points. These results contain some previous results of Kahane and Taylor-Williams obtained independently.
On désigne par une algèbre de Banach homogène sur le cercle et par la sous-algèbre fermée de constituée par les fonctions qui ont des prolongements analytiques dans le disque ouvert . Ce travail considère la structure des idéaux fermés de , sous des restrictions convenables sur les propriétés de synthèse de . En particulier, on caractérise complètement les idéaux fermés de tels que les “zero sets” rencontrent le cercle en un ensemble dénombrable. Ces résultats contiennent des résultats précédents de Kahane et de Taylor-Williams obtenus indépendamment.
@article{AIF_1972__22_3_1_0, author = {Bennett, Colin and Gilbert, John E.}, title = {Homogeneous algebras on the circle. {I.} {Ideals} of analytic functions}, journal = {Annales de l'Institut Fourier}, pages = {1--19}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {22}, number = {3}, year = {1972}, doi = {10.5802/aif.422}, mrnumber = {49 #3546}, zbl = {0228.46046}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.422/} }
TY - JOUR AU - Bennett, Colin AU - Gilbert, John E. TI - Homogeneous algebras on the circle. I. Ideals of analytic functions JO - Annales de l'Institut Fourier PY - 1972 SP - 1 EP - 19 VL - 22 IS - 3 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/aif.422/ DO - 10.5802/aif.422 LA - en ID - AIF_1972__22_3_1_0 ER -
%0 Journal Article %A Bennett, Colin %A Gilbert, John E. %T Homogeneous algebras on the circle. I. Ideals of analytic functions %J Annales de l'Institut Fourier %D 1972 %P 1-19 %V 22 %N 3 %I Institut Fourier %C Grenoble %U http://archive.numdam.org/articles/10.5802/aif.422/ %R 10.5802/aif.422 %G en %F AIF_1972__22_3_1_0
Bennett, Colin; Gilbert, John E. Homogeneous algebras on the circle. I. Ideals of analytic functions. Annales de l'Institut Fourier, Volume 22 (1972) no. 3, pp. 1-19. doi : 10.5802/aif.422. http://archive.numdam.org/articles/10.5802/aif.422/
[1] Questions of regularity connected with the Phragmen-Lindelöf principle, Ann. of Math., (2) 50 (1949), 341-346. | MR | Zbl
and ,[2] On the Harmonic Analysis of Rearrangement-Invariant Banach Function Spaces, Thesis, University of Newcastle, 1971.
,[3] “Entire Functions”, Academic Press (1954), New York. | MR | Zbl
,[4] “L'intégrale de Fourier et questions qui s'y rattachent”, Almqvist and Wiksell (1944), Uppsala. | Zbl
,[5] On the existence of a largest subharmonic minorant of a given function, Ark. Mat. 3, (1967), 429-440. | MR | Zbl
,[6] On the harmonic analysis of some subalgebras of L1 (0, ∞), Seminar, Symposium in Harmonic Analysis, Warwick (1968).
,[7] On primary ideals in the space L1 (0, ∞), Soviet Math. Doklady, 7 (1966), 266-268. | Zbl
,[8] Invariant subspaces of analytic functions, Canad. J. Math., 17 (1965), 643-651. | MR | Zbl
and ,[9] “Banach spaces of analytic functions”, Prentice-Hall (1962), Englewood Cliffs, N.J. | MR | Zbl
,[10] Idéaux primaires fermés dans certaines algèbres de Banach de fonctions analytiques, (to appear). | Zbl
,[11] “An introduction to harmonic analysis”, John Wiley (1968), New York. | MR | Zbl
,[12] On the spectral analysis of bounded functions, Pacific J. Math., 16 (1966), 121-128. | MR | Zbl
,[13] A generalization of Wiener's Tauberian Theorem and spectrum of fast growing functions, Trudy Moskov Mat. Obsc., 7 (1958), 121-148.
,[14] Homogeneous algebras on compact abelian groups, Trans. Amer. Math. Soc., 87 (1958), 372-386. | MR | Zbl
,[15] The work of Silov on commutative Banach algebras, Notas de Matematica, No. 20 (1959), Rio de Janeiro. | MR | Zbl
,[16] On the one-dimensional translation group and semi-group in certain function spaces, Thesis (1950), Uppsala. | MR | Zbl
,[17] Continuous functions on compact spaces without perfect subsets, Proc. Amer. Math. Soc., 8 (1957), 39-42. | MR | Zbl
,[18] Homogeneous rings of functions, Amer. Math. Soc. Translation No. 92. | Zbl
,[19] The space of functions analytic in the disk with indefinitely differentiable boundary values, (submitted for publication).
and ,Cited by Sources: