Illinois Journal of Mathematics

On the spectrum of Banach algebra-valued entire functions

J. P. Bannon, P. Cade, and R. Yang

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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.

Article information

Illinois J. Math., Volume 55, Number 4 (2011), 1455-1465.

First available in Project Euclid: 12 July 2013

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Zentralblatt MATH identifier

Primary: 32A65: Banach algebra techniques [See mainly 46Jxx]
Secondary: 47A10: Spectrum, resolvent


Bannon, J. P.; Cade, P.; Yang, R. On the spectrum of Banach algebra-valued entire functions. Illinois J. Math. 55 (2011), no. 4, 1455--1465. doi:10.1215/ijm/1373636693.

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