Homogeneous algebras on the circle. I. Ideals of analytic functions
Annales de l'Institut Fourier, Volume 22 (1972) no. 3, p. 1-19

Let 𝒜 be a homogeneous algebra on the circle and 𝒜 + the closed subalgebra of 𝒜 of functions having analytic extensions into the unit disk D. This paper considers the structure of closed ideals of 𝒜 + under suitable restrictions on the synthesis properties of 𝒜. In particular, completely characterized are the closed ideals in 𝒜 + 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 𝒜 une algèbre de Banach homogène sur le cercle et par 𝒜 + la sous-algèbre fermée de 𝒜 constituée par les fonctions qui ont des prolongements analytiques dans le disque ouvert D. Ce travail considère la structure des idéaux fermés de 𝒜 + , sous des restrictions convenables sur les propriétés de synthèse de 𝒜. En particulier, on caractérise complètement les idéaux fermés de 𝒜 + 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},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {22},
     number = {3},
     year = {1972},
     pages = {1-19},
     doi = {10.5802/aif.422},
     zbl = {0228.46046},
     mrnumber = {49 \#3546},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/AIF_1972__22_3_1_0/

[1] L. Ahlfors and M. Heins, Questions of regularity connected with the Phragmen-Lindelöf principle, Ann. of Math., (2) 50 (1949), 341-346. | MR 10,522c | Zbl 0036.04702

[2] C. Bennett, On the Harmonic Analysis of Rearrangement-Invariant Banach Function Spaces, Thesis, University of Newcastle, 1971.

[3] R.P. Boas, “Entire Functions”, Academic Press (1954), New York. | MR 16,914f | Zbl 0058.30201

[4] T. Carleman, “L'intégrale de Fourier et questions qui s'y rattachent”, Almqvist and Wiksell (1944), Uppsala. | Zbl 0060.25504

[5] Y. Domar, On the existence of a largest subharmonic minorant of a given function, Ark. Mat. 3, (1967), 429-440. | MR 19,408c | Zbl 0078.09301

[6] J.E. Gilbert, On the harmonic analysis of some subalgebras of L1 (0, ∞), Seminar, Symposium in Harmonic Analysis, Warwick (1968).

[7] V.P. Gurarii, On primary ideals in the space L1 (0, ∞), Soviet Math. Doklady, 7 (1966), 266-268. | Zbl 0154.39203

[8] M. Hasumi and T. Srinivasan, Invariant subspaces of analytic functions, Canad. J. Math., 17 (1965), 643-651. | MR 31 #3871 | Zbl 0128.34201

[9] K. Hoffman, “Banach spaces of analytic functions”, Prentice-Hall (1962), Englewood Cliffs, N.J. | MR 24 #A2844 | Zbl 0117.34001

[10] J.P. Kahane, Idéaux primaires fermés dans certaines algèbres de Banach de fonctions analytiques, (to appear). | Zbl 0271.46047

[11] Y. Katznelson, “An introduction to harmonic analysis”, John Wiley (1968), New York. | MR 40 #1734 | Zbl 0169.17902

[12] P. Koosis, On the spectral analysis of bounded functions, Pacific J. Math., 16 (1966), 121-128. | MR 32 #8044 | Zbl 0146.12201

[13] B.I. Korenblyum, A generalization of Wiener's Tauberian Theorem and spectrum of fast growing functions, Trudy Moskov Mat. Obsc., 7 (1958), 121-148.

[14] K. De Leeuw, Homogeneous algebras on compact abelian groups, Trans. Amer. Math. Soc., 87 (1958), 372-386. | MR 21 #820 | Zbl 0083.34603

[15] H. Mirkil, The work of Silov on commutative Banach algebras, Notas de Matematica, No. 20 (1959), Rio de Janeiro. | MR 33 #3130 | Zbl 0090.09301

[16] B. Nyman, On the one-dimensional translation group and semi-group in certain function spaces, Thesis (1950), Uppsala. | MR 12,108g | Zbl 0037.35401

[17] W. Rudin, Continuous functions on compact spaces without perfect subsets, Proc. Amer. Math. Soc., 8 (1957), 39-42. | MR 19,46b | Zbl 0077.31103

[18] G.E. Silov, Homogeneous rings of functions, Amer. Math. Soc. Translation No. 92. | Zbl 0053.08401

[19] B.A. Taylor and D.L. Williams, The space of functions analytic in the disk with indefinitely differentiable boundary values, (submitted for publication).