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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
Article

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 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
.

Keywords

Continuous Function Open Disk Nontangential Limit 
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Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

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

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