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

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## Abstract

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 algebra*B*. In this paper we describe the space*M*(H_{ B }^{ G }) of maximal ideals of the algebraH_{ B }^{ G }and prove that*M*(H_{ B }^{ G }) =*M*(*B*) ∪*M*(H_{ B }^{ G }and prove that*M*(H_{0}^{ G }), whereH_{0}^{ G }is a closed ideal in*G*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 algebra*B*, where*I B*= {*u*_{α}}_{ B }are inner functions such that\(\bar u_\alpha \in B\)on.## Keywords

Continuous Function Open Disk Nontangential Limit
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## References

- 1.J. B. Garnett,
*Bounded Analytical Functions*, Academic Press, New York (1981).Google Scholar - 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.P. Gorkin and K. Izuchi, “Some counterexamples in subalgebras of
*L*^{∞}(*D*),”*Indiana Univ. Math. J.*,**40**, 1301–1313 (1991).Google Scholar - 4.S. Axler and A. Shields, “Algebras generated by analytic and harmonic functions,”
*Indiana Univ. Math. J.*,**36**, 631–638 (1987).Google Scholar - 5.C. Bishop,
*A Characterization of Some Algebras on the Disk*, Preprint (1992).Google Scholar - 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.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.K. Hoffman, “Bounded analytical functions and Gleason parts,”
*Ann. Math.*,**86**, 74–111 (1967).Google Scholar - 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.O. V. Ivanov, “Generalized analytic functions and analytic subalgebras,”
*Ukr. Mat. Zh.*,**42**, 616–620 (1990).Google Scholar - 11.O. V. Ivanov, “Generalized Douglas algebras and Corona theorem,”
*Sib. Mat. Zh.*,**32**, 37–42 (1991).Google Scholar - 12.O. V. Ivanov, “Nontangential limits and Shilov boundary of the algebra
*H*^{∞},”*Dokl. Akad. Nauk Ukr. SSR, Ser. A*, No. 7, 5–8 (1991).Google Scholar - 13.S. Axler and A. Shields, “Extensions of harmonic and analytic functions,”
*Pac. J. Math.*,**145**, No. 1, 1–15 (1990).Google Scholar - 14.
- 15.M. Naimark,
*Normed Rings*[in Russian], Nauka, Moscow (1968).Google Scholar - 16.T. Gamelin,
*Uniform Algebras*, Prentice-Hall (1969).Google Scholar - 17.
- 18.L. Brown and P. M. Gauthier, “Behavior of normal meromorphic function on the maximal ideal space of
*H*^{∞},”*Mich. Math. J.*,**18**, 365–371 (1971).Google Scholar

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© Plenum Publishing Corporation 1994