Abstract
LetH G B 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 G B ) of maximal ideals of the algebraH G B and prove thatM(H G B ) =M(B) ∪M(H G B and prove thatM(H G0 ), whereH G0 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 G B . We also prove that\(\mathcal{H}_B^G = alg (\mathcal{H}_{H^\infty }^G ,\overline {IB} )\) for any Douglas algebraB, whereI B = {u α} B are inner functions such that\(\bar u_\alpha \in B\)on
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References
J. B. Garnett,Bounded Analytical Functions, Academic Press, New York (1981).
S. Axler and P. Gorkin, “Algebras on the disk and doubly commuting multiplication operators,”Trans. Am. Math. Soc.,309, 711–723 (1988).
P. Gorkin and K. Izuchi, “Some counterexamples in subalgebras ofL ∞(D),”Indiana Univ. Math. J.,40, 1301–1313 (1991).
S. Axler and A. Shields, “Algebras generated by analytic and harmonic functions,”Indiana Univ. Math. J.,36, 631–638 (1987).
C. Bishop,A Characterization of Some Algebras on the Disk, Preprint (1992).
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).
O. V. Ivanov,The Axler-Shields Problem of Nontangential Limits and a Maximal Ideal Space of Some Pseudoanalytic Algebras, Preprint (1992), pp. 1–3.
K. Hoffman, “Bounded analytical functions and Gleason parts,”Ann. Math.,86, 74–111 (1967).
O. V. Ivanov, “Generalized analytic functions and 2-sheeted Corona theorem,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 4, 10–11 (1989).
O. V. Ivanov, “Generalized analytic functions and analytic subalgebras,”Ukr. Mat. Zh.,42, 616–620 (1990).
O. V. Ivanov, “Generalized Douglas algebras and Corona theorem,”Sib. Mat. Zh.,32, 37–42 (1991).
O. V. Ivanov, “Nontangential limits and Shilov boundary of the algebraH ∞,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 7, 5–8 (1991).
S. Axler and A. Shields, “Extensions of harmonic and analytic functions,”Pac. J. Math.,145, No. 1, 1–15 (1990).
T. Gamelin,Lectures on H ∞(D), Universidad National de La Plata, La Plata (1972).
M. Naimark,Normed Rings [in Russian], Nauka, Moscow (1968).
T. Gamelin,Uniform Algebras, Prentice-Hall (1969).
C. Sundberg, “Truncation of B.MO function,”Indiana Univ. Math. J.,33, 749–771 (1984).
L. Brown and P. M. Gauthier, “Behavior of normal meromorphic function on the maximal ideal space ofH ∞,”Mich. Math. J.,18, 365–371 (1971).
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Published in Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 924–931, July, 1993.
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Ivanov, O.V. The structure of Banach algebras of bounded continuous functions on the open disk that contain H∞, the Hoffman algebra, and nontangential limits. Ukr Math J 45, 1023–1030 (1993). https://doi.org/10.1007/BF01057449
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DOI: https://doi.org/10.1007/BF01057449