Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Countably generated Douglas algebras


Author: Keiji Izuchi
Journal: Trans. Amer. Math. Soc. 299 (1987), 171-192
MSC: Primary 46J15; Secondary 30D55, 30H05
DOI: https://doi.org/10.1090/S0002-9947-1987-0869406-9
MathSciNet review: 869406
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Under a certain assumption of $ f$ and $ g$ in $ {L^\infty }$ which is considered by Sarason, a strong separation theorem is proved. This is available to study a Douglas algebra $ [{H^\infty },\,f]$ generated by $ {H^\infty }$ and $ f$. It is proved that (1) ball $ (B/{H^\infty } + C)$ does not have exposed points for every Douglas algebra $ B$, (2) Sarason's three functions problem is solved affirmatively, (3) some characterization of $ f$ for which $ [{H^\infty },\,f]$ is singly generated, and (4) the $ M$-ideal conjecture for Douglas algebras is not true.


References [Enhancements On Off] (What's this?)

  • [1] S. Axler, Factorization of $ {L^\infty }$ functions, Ann. of Math. (2) 106 (1977), 567-572. MR 0461142 (57:1127)
  • [2] S. Axler, I. D. Berg, N. Jewell, and A. L. Shields, Approximation by compact operators and the space $ {H^\infty } + C$, Ann. of Math. (2) 109 (1979), 601-612. MR 534765 (81h:30053)
  • [3] S.-Y. A. Chang, A characterization of Douglas subalgebras, Acta Math. 137 (1976), 81-89. MR 0428044 (55:1074a)
  • [4] T. Gamelin, Uniform algebras, Prentice-Hall, Englewood Cliffs, N.J., 1969. MR 0410387 (53:14137)
  • [5] T. Gamelin, D. Marshall, R. Younis, and W. Zame, Function theory and $ M$-ideals, Ark. Mat. 23 (1985), 261-279. MR 827346 (87m:46102)
  • [6] J. Garnett, Bounded analytic functions, Academic Press, New York and London, 1981. MR 628971 (83g:30037)
  • [7] P. Gorkin, Decompositions of the maximal ideal space of $ {L^\infty }$, Thesis, Michigan State Univ., East Lansing, 1982.
  • [8] -, Decompositions of the maximal ideal space of $ {L^\infty }$, Trans. Amer. Math. Soc. 282 (1984), 33-44. MR 728701 (85a:46028)
  • [9] C. Guillory, K. Izuchi and D. Sarason, Interpolating Blaschke products and division in Douglas algebras, Proc. Roy. Irish Acad. 84A (1984), 1-7. MR 771641 (86j:46054)
  • [10] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, Englewood Cliffs, N.J., 1962. MR 0133008 (24:A2844)
  • [11] -, unpublished note.
  • [12] K. Izuchi, Zero sets of interpolating Blaschke products, Pacific J. Math. 119 (1985), 337-342. MR 803123 (87i:30069)
  • [13] -, $ QC$-level sets and quotients of Douglas algebras, J. Funct. Anal. 65 (1986), 293-308. MR 826428 (87f:46093)
  • [14] -, A geometrical characterization of singly generated Douglas algebras, Proc. Amer. Math. Soc. 97 (1986), 410-412. MR 840620 (87e:46068)
  • [15] K. Izuchi and Y. Izuchi, Extreme and exposed points in quotients of Douglas algebras by $ {H^\infty }$ or $ {H^\infty } + C$, Yokohama Math. J. 32 (1984), 45-54. MR 772904 (86d:46045)
  • [16] -, Annihilating measures for Douglas algebras, Yokohama Math. J. 32 (1984), 135-151. MR 772911 (86e:46042)
  • [17] D. Luecking, The compact Hankel operators from an $ M$-ideal in the space of Hankel operators, Proc. Amer. Math. Soc. 79 (1980), 222-224. MR 565343 (81h:46057)
  • [18] D. Luecking and R. Younis, Quotients of $ {L^\infty }$ by Douglas algebras and best approximation, Trans. Amer. Math. Soc. 276 (1983), 699-706. MR 688971 (84f:46074)
  • [19] D. Marshall, Subalgebras of $ {L^\infty }$ containing $ {H^\infty }$, Acta Math. 137 (1976), 91-98. MR 0428045 (55:1074b)
  • [20] D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405. MR 0377518 (51:13690)
  • [21] -, Function theory on the unit circle, Virginia Polytechnic Institute and State Univ., Blacksburg, 1978. MR 521811 (80d:30035)
  • [22] -, The Shilov and Bishop decompositions of $ {H^\infty } + C$, Conf. on Harmonic Anal. in Honor of A. Zygmund, Vol. 2, Wadsworth, Belmont, Calif., 1981, pp. 461-474. MR 730085 (86a:46064)
  • [23] T. Wolff, Two algebras of bounded functions, Duke Math. J. 49 (1982), 321-328. MR 659943 (84c:30051)
  • [24] R. Younis, Division in Douglas algebras and some applications, Arch. Math. (Basel) 45 (1985), 550-560. MR 818297 (87b:46059)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46J15, 30D55, 30H05

Retrieve articles in all journals with MSC: 46J15, 30D55, 30H05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1987-0869406-9
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society