Solvability of the B\'ezout Equation for Banach Algebra-Valued $H^\infty$ Functions on the Polydisk

TL;DR

This paper investigates the conditions under which the Bézout equation can be solved for Banach algebra-valued bounded holomorphic functions on the polydisk, contributing to the understanding of the multidimensional corona problem.

Contribution

It introduces a method to extend local solutions of the Bézout equation to global solutions using a dimension-induction scheme and topological analysis of the maximal ideal space.

Findings

Established solvability conditions for the Bézout equation in broader subalgebras.

Connected local solvability to global solutions via topological methods.

Extended previous results to new classes of subalgebras.

Abstract

In connection with the still unsolved multidimensional corona problem for algebras of bounded holomorphic functions on convex domains, we study the solvability of the B\'ezout equation for the algebra of bounded holomorphic functions on the polydisk with values in a complex Banach algebra. Assuming local solvability of the B\'ezout equation on a special open cover of the maximal ideal space of the algebra, we combine a dimension-induction scheme with a careful analysis of the topological structure of this space to glue local solutions into a global one. As a corollary, we obtain the solvability of the B\'ezout equation for a broader class of subalgebras containing the slice algebra of bounded holomorphic functions, the case of the latter having been previously proved by the first author