Topology of the Maximal Ideal Space of H^∞ Revisited

2015/07/14 by Alexander Brudnyi, Brudnyi, Alexander
Mathematics · #30D55 #30H05 #Advanced Operator Algebra Research #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1507.03669

openalex publication_date 2015/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M(H^∞) be the maximal ideal space of the Banach algebra H^∞ of bounded holomorphic functions on the unit disk \mathbb D⊂\mathbb C. We prove that M(H^∞) is homeomorphic to the Freudenthal compactification γ(M_a) of the set M_a of all non-trivial (analytic disks) Gleason parts of M(H^∞). Also, we give alternative proofs of important results of Suárez asserting that the set M_s of trivial (one-pointed) Gleason parts of M(H^∞) is totally disconnected and that the Čech cohomology group H²(M(H^∞),\mathbb Z)=0.

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