Geodesic
On the Extreme Points of Quotients of L∞ by Douglas Algebras
Canadian mathematical bulletin, Tome 27 (1984) no. 4, pp. 517-522

Voir la notice de l'article provenant de la source Cambridge University Press

Let B be a Douglas algebra which admits best approximation. It will be shown that the following are equivalent: (1) The unit ball of (L∞/B) has no extreme points; (2) For any Blaschke product b with , there exists h ∈ B such that = 1 and h|E≢0, where E is the essential set of B.It will also be proven that if B⊇H∞+C and its essential set E contains a closed Gδ set, then the unit ball of (L∞/B) has no extreme points. Many known results concerning this subject will follow from these results.
DOI : 10.4153/CMB-1984-083-0
Mots-clés : 30H05, 46E15, 41A50 
Deeb, Waleed; Younis, Rahman. On the Extreme Points of Quotients of L∞ by Douglas Algebras. Canadian mathematical bulletin, Tome 27 (1984) no. 4, pp. 517-522. doi: 10.4153/CMB-1984-083-0
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[1] 1. Axler, S., Berg, I., Jewell, N. and Shields, A., Approximation by compact operators and the space H∞+C, Annals of Math., 109(1979), 601-612.10.2307/1971228 Google Scholar | DOI

[2] 2. Clancey, K. F. and Gosselin, J. A., On the local theory of Toeplitz operators, Illinois J. Math. 22 (1978), 449-458.10.1215/ijm/1256048607 Google Scholar | DOI

[3] 3. Davie, A. M., Gamelin, T. W. and Garnett, J., Distance estimates and point-wise bounded density, Trans. Aoner. Math. Soc. 175(1973), 37-68. Google Scholar | DOI

[4] 4. Gamelin, T. W., Uniform algebras, Prentice-Hall (1969). Google Scholar

[5] 5. Garnett, J., Bounded analytic functions, Academic Press (1981). Google Scholar

[6] 6. Gorkin, P., Decomposition of the maximal ideal space of L∞, Ph.D. -thesis, Michigan State University, 1982. Google Scholar

[7] 7. Hoffman, K., Banach spaces of analytic functions, Prentice-Hall (1962). Google Scholar

[8] 8. Izuchi, K. and Younis, R., On the quotient space of two Douglas algebras, submitted. Google Scholar

[9] 9. Izuchi, K., Extreme points of unit balls of quotients of L∞ by Douglas algebras, preprint.10.1215/ijm/1256044756 Google Scholar | DOI

[10] 10. Koosis, P., Weighted quadratic means of Hilbert transforms, Duke Math. J. 38 (1971), 609-634.10.1215/S0012-7094-71-03877-4 Google Scholar | DOI

[11] 11. Leucking, D. and Younis, R., Quotients of L∞ by Douglas algebras and Best approximation, Transaction, Amer. Math. Soc. V276(1983), 699-706. Google Scholar

[12] 12. Sarson, D., Function theory on the unit circle, Virginia Poly. Inst, and State Univ. (1979). Google Scholar

[13] 13. Sundberg, C., A note on algebras between L∞ and H∞ , Rocky Mount. J. Math. 11(1981), 333-336. 10.1216/RMJ-1981-11-2-333 Google Scholar | DOI

[14] 14. Younis, R., Best approximation in certain Douglas algebras, Proc. Amer. Math. Soc. 80 (1980), 639-642.10.1090/S0002-9939-1980-0587943-0 Google Scholar | DOI

[15] 15. Younis, R., Properties of certain algebras between L∞ and H∞, J. Funct. Anal. 44 (1981), 381-387.10.1016/0022-1236(81)90016-1 Google Scholar | DOI

[16] 16. Younis, R., M-ideals of L∞/H∞ and support sets, to appear in Illinois J. Math. Google Scholar

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