On a class of inner maps

Vesentini, Edoardo. "On a class of inner maps." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 16.4 (2005): 215-226. <http://eudml.org/doc/252420>.

@article{Vesentini2005,
abstract = { Let $f$ be a continuous map of the closure $\overline\{\Delta\}$ of the open unit disc $\Delta$ of $\mathbb\{C\}$ into a unital associative Banach algebra $\mathcal\{A\}$, whose restriction to $\Delta$ is holomorphic, and which satisfies the condition whereby $0 \notin \sigma(f(z)) \subset \overline\{\Delta\}$ for all $z \in \Delta$ and $\sigma(f(z)) \subset \partial \Delta$ whenever $z \in \partial \Delta$ (where $\sigma(x)$ is the spectrum of any $x \in \mathcal\{A\}$). One of the basic results of the present paper is that $f$ is , that is to say, $\sigma(f(z))$ is then a compact subset of $\partial \Delta$ that does not depend on $z$ for all $z \in \overline\{\Delta\}$. This fact will be applied to holomorphic self-maps of the open unit ball of some $J^\{*\}$-algebra and in particular of any unital $C^\{*\}$-algebra, investigating some cases in which not only the spectra but the maps themselves are necessarily constant.},
author = {Vesentini, Edoardo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Associative Banach algebra; Holomorphic map; Spectrum; Spectral radius; associative Banach space; holomorphic map; spectrum; spectral radius},
language = {eng},
month = {12},
number = {4},
pages = {215-226},
publisher = {Accademia Nazionale dei Lincei},
title = {On a class of inner maps},
url = {http://eudml.org/doc/252420},
volume = {16},
year = {2005},
}

TY - JOUR
AU - Vesentini, Edoardo
TI - On a class of inner maps
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2005/12//
PB - Accademia Nazionale dei Lincei
VL - 16
IS - 4
SP - 215
EP - 226
AB - Let $f$ be a continuous map of the closure $\overline{\Delta}$ of the open unit disc $\Delta$ of $\mathbb{C}$ into a unital associative Banach algebra $\mathcal{A}$, whose restriction to $\Delta$ is holomorphic, and which satisfies the condition whereby $0 \notin \sigma(f(z)) \subset \overline{\Delta}$ for all $z \in \Delta$ and $\sigma(f(z)) \subset \partial \Delta$ whenever $z \in \partial \Delta$ (where $\sigma(x)$ is the spectrum of any $x \in \mathcal{A}$). One of the basic results of the present paper is that $f$ is , that is to say, $\sigma(f(z))$ is then a compact subset of $\partial \Delta$ that does not depend on $z$ for all $z \in \overline{\Delta}$. This fact will be applied to holomorphic self-maps of the open unit ball of some $J^{*}$-algebra and in particular of any unital $C^{*}$-algebra, investigating some cases in which not only the spectra but the maps themselves are necessarily constant.
LA - eng
KW - Associative Banach algebra; Holomorphic map; Spectrum; Spectral radius; associative Banach space; holomorphic map; spectrum; spectral radius
UR - http://eudml.org/doc/252420
ER -