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

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 $\mathbb C^n$.