On the spectrum of Banach algebra-valued entire functions

Abstract

In this paper, we investigate a notion of spectrum $\sigma(f)$ for Banach algebra-valued holomorphic functions on $\mathbb{C}^{n}$. We prove that the resolvent $\sigma^{c}(f)$ is a disjoint union of domains of holomorphy when $\mathcal{B}$ is a $C^{\ast}$-algebra or is reflexive as a Banach space. Further, we study the topology of the resolvent via consideration of the $\mathcal{B}$-valued Maurer–Cartan type $1$-form $f(z)^{-1}\,df(z)$. As an example, we explicitly compute the spectrum of a linear function associated with the tuple of standard unitary generators in a free group factor von Neumann algebra.