On a theorem of Vesentini
Volume 162 / 2004
Studia Mathematica 162 (2004), 183-193
MSC: Primary 46H30.
DOI: 10.4064/sm162-2-6
Abstract
Let ${\mathcal A}$ be a Banach algebra over ${\mathbb C}$ with unit ${\bf 1}$ and $f: {\mathbb C} \to {\mathbb C}$ an entire function. Let ${\bf f}: {\mathcal A} \to {\mathcal A}$ be defined by $$ {\bf f} (a)={f}(a) \hskip 1em (a\in {\mathcal A}), $$ where $f(a)$ is given by the usual analytic calculus. The connections between the periods of $f$ and the periods of ${\bf f}$ are settled by a theorem of E. Vesentini. We give a new proof of this theorem and investigate further properties of periods of ${\bf f}$, for example in $C^\ast $-algebras.
