Ukrainian Mathematical Journal

, Volume 45, Issue 7, pp 1023–1030 | Cite as

The structure of Banach algebras of bounded continuous functions on the open disk that contain H, the Hoffman algebra, and nontangential limits

  • O. V. Ivanov


LetH B G be an algebra of bounded continuous functions on an open disk
representable in the formH B G, where\(G \mathop = \limits^{def} C(M(H^\infty )) = alg (H^\infty ,\overline H ^\infty )\) andH B is a closed subalgebra in C(D) consisting of the functions that have nontangential limits almost everywhere on {ie1023-06}, and these limits belong to the Douglas algebraB. In this paper we describe the spaceM(H B G ) of maximal ideals of the algebraH B G and prove thatM(H B G ) =M(B) ∪M(H B G and prove thatM(H 0 G ), whereH 0 G is a closed ideal inG consisting of functions having nontangential limits equal to zero almost everywhere on {ie1023-12}. Moreover, it is established that\(H^{\infty \supset } [\overline Z ] \ne \mathcal{H}_{H^\infty + C}^G \) on the disk. The Chang-Marshall theorem is generalized for the Banach algebrasH B G . We also prove that\(\mathcal{H}_B^G = alg (\mathcal{H}_{H^\infty }^G ,\overline {IB} )\) for any Douglas algebraB, whereI B = {uα} B are inner functions such that\(\bar u_\alpha \in B\)on


Continuous Function Open Disk Nontangential Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. B. Garnett,Bounded Analytical Functions, Academic Press, New York (1981).Google Scholar
  2. 2.
    S. Axler and P. Gorkin, “Algebras on the disk and doubly commuting multiplication operators,”Trans. Am. Math. Soc.,309, 711–723 (1988).Google Scholar
  3. 3.
    P. Gorkin and K. Izuchi, “Some counterexamples in subalgebras ofL (D),”Indiana Univ. Math. J.,40, 1301–1313 (1991).Google Scholar
  4. 4.
    S. Axler and A. Shields, “Algebras generated by analytic and harmonic functions,”Indiana Univ. Math. J.,36, 631–638 (1987).Google Scholar
  5. 5.
    C. Bishop,A Characterization of Some Algebras on the Disk, Preprint (1992).Google Scholar
  6. 6.
    O. V. Ivanov, “The Fatou theorem on nontangential limits and problems of extension on the ideal boundary,”Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov.,19, 101–109 (1991).Google Scholar
  7. 7.
    O. V. Ivanov,The Axler-Shields Problem of Nontangential Limits and a Maximal Ideal Space of Some Pseudoanalytic Algebras, Preprint (1992), pp. 1–3.Google Scholar
  8. 8.
    K. Hoffman, “Bounded analytical functions and Gleason parts,”Ann. Math.,86, 74–111 (1967).Google Scholar
  9. 9.
    O. V. Ivanov, “Generalized analytic functions and 2-sheeted Corona theorem,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 4, 10–11 (1989).Google Scholar
  10. 10.
    O. V. Ivanov, “Generalized analytic functions and analytic subalgebras,”Ukr. Mat. Zh.,42, 616–620 (1990).Google Scholar
  11. 11.
    O. V. Ivanov, “Generalized Douglas algebras and Corona theorem,”Sib. Mat. Zh.,32, 37–42 (1991).Google Scholar
  12. 12.
    O. V. Ivanov, “Nontangential limits and Shilov boundary of the algebraH ,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 7, 5–8 (1991).Google Scholar
  13. 13.
    S. Axler and A. Shields, “Extensions of harmonic and analytic functions,”Pac. J. Math.,145, No. 1, 1–15 (1990).Google Scholar
  14. 14.
    T. Gamelin,Lectures on H (D), Universidad National de La Plata, La Plata (1972).Google Scholar
  15. 15.
    M. Naimark,Normed Rings [in Russian], Nauka, Moscow (1968).Google Scholar
  16. 16.
    T. Gamelin,Uniform Algebras, Prentice-Hall (1969).Google Scholar
  17. 17.
    C. Sundberg, “Truncation of B.MO function,”Indiana Univ. Math. J.,33, 749–771 (1984).Google Scholar
  18. 18.
    L. Brown and P. M. Gauthier, “Behavior of normal meromorphic function on the maximal ideal space ofH ,”Mich. Math. J.,18, 365–371 (1971).Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • O. V. Ivanov
    • 1
  1. 1.Institute of Applied Mathematics and MechanicsUkrainian Academy of SciencesDonetsk

Personalised recommendations