Reducibility of invertible tuples to the principal component in commutative Banach algebras

TL;DR

This paper explores conditions under which invertible tuples in commutative Banach algebras can be reduced to the principal component, extending classical results with new analytical and topological insights.

Contribution

It introduces two notions of exponential reducibility and establishes a link between reducibility in the algebra and its maximal ideal space, with geometric and topological criteria.

Findings

Established necessary and sufficient conditions for reducibility.

Extended the Corach-Suárez result to new settings.

Provided analytical methods for reducibility analysis.

Abstract

Let $A$ be a complex, commutative unital Banach algebra. We introduce two notions of exponential reducibility of Banach algebra tuples and present an analogue to the Corach-Su\'arez result on the connection between reducibility in $A$ and in $C(M(A))$. Our methods are of an analytical nature. Necessary and sufficient geometric/topological conditions are given for reducibility (respectively reducibility to the principal component of $U_n(A)$) whenever the spectrum of $A$ is homeomorphic to a subset of $\mathbb C^n$.