General solution of corona problem

2025/05/26 by Marek Kosiek, Kosiek, Marek, Krzysztof Rudol +1
Engineering · #32A65 #46J15 #46J20 #Aerosol Filtration and Electrostatic Precipitation #FOS: Mathematics #Functional Analysis (math.FA) #Primary 30H80 #Secondary 32A38

paper · pdf · doi:10.48550/arxiv.2505.20220

openalex publication_date 2025/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using a description of the spectrum of bidual algebra A^** of a uniform algebra A we obtain abstract corona theorem for certain uniform algebras. It asserts the density of a specific Gleason part in the spectrum of an H^∞ -- type subalgebra of A^**. There is an isometric isomorphism of the latter subalgebra with H^∞(G) for a wide class of domains G⊂\mathbb C^d. Using abstract corona theorem we show the density of the canonical image of G in the spectrum of H^∞(G), solving positively corona problem for such domains. In particular, we obtain positive solution for balls and polydisks.

Related

• Corona Theorem
• Estimates for the Corona Theorem on H^∞_\mathbbI(\D)
• Another proof of the corona theorem
• Obstructions to countable saturation in corona algebras
• A new proof of the corona problem