The Corona Problem for Slice Hyperholomorphic Functions

TL;DR

This paper solves the Corona problem for slice hyperholomorphic functions over quaternions by reformulating the problem and adapting classical proofs, extending complex analysis techniques into hypercomplex algebra.

Contribution

It introduces a novel approach to the quaternionic Corona problem by reformulating the Bezout equation and adapting Wolff's proof within the hypercomplex setting.

Findings

Resolved the quaternionic Corona problem in the slice hyperholomorphic setting.

Reformulated the Bezout equation for quaternionic functions.

Extended classical complex analysis methods to hypercomplex algebra.

Abstract

This paper addresses the Corona problem for slice hyperholomorphic functions for a single quaternionic variable. While the Corona problem is well-understood in the context of one complex variable, it remains highly challenging in the case of several complex variables. The extension of the theory of one complex variable to several complex variables is not the only possible extension to multi-dimensional complex analysis. Instead of functions holomorphic in each variable separately, in this paper we will consider functions in some hypercomplex algebras, in particular in the algebra of quaternions. Previously, the Corona problem had not been studied within the hypercomplex framework because of challenges posed by pointwise multiplication, which is not closed for hypercomplex-valued analytic functions. Alternative notions of multiplication that are closed often compromise other desirable properties, further complicating the analysis. In this work, we resolve the Corona problem within the quaternionic slice hyperholomorphic setting. Our approach involves reformulating the quaternionic Bezout equation with respect to the appropriate multiplication into a new system of Bezout equations on the unit disc. We solve this system by adapting Wolff's proof of the Corona theorem for bounded analytic functions. As the number of generators increases, the associated algebra grows increasingly intricate.