Wallman-type compaerifications and function lattices

Caterino, Alessandro, and Vipera, Maria Cristina. "Wallman-type compaerifications and function lattices." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 82.4 (1988): 679-683. <http://eudml.org/doc/287308>.

@article{Caterino1988,
abstract = {Let $F \subset C^\{\ast\} (X)$ be a vector sublattice over $\mathbb\{R\}$ which separates points from closed sets of $X$. The compactification $e_\{F\}X$ obtained by embedding $X$ in a real cube via the diagonal map, is different, in general, from the Wallman compactification $\omega (Z(F))$. In this paper, it is shown that there exists a lattice $F_\{z\}$ containing $F$ such that $\omega (Z(F)) = \omega (Z(F_\{z\})) = e_\{F\}X$. In particular this implies that $\omega (Z(F)) \ge e_\{F\}X$. Conditions in order to be $\omega (Z(F)) = e_\{F\}X$ are given. Finally we prove that, if $\alpha X$ is a compactification of $X$ such that $Cl_\{\alpha X\} (\alpha X \setminus X)$ is $0$-dimensional, then there is an algebra $A \subset C^\{ast\} (X)$ such that $\omega (Z(A)) = e_\{A\} X = \alpha X$.},
author = {Caterino, Alessandro, Vipera, Maria Cristina},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Compactifications; Normal bases; Function lattices; Zero-sets; normal bases; function lattices; zero-sets; Wallman compactification},
language = {eng},
month = {12},
number = {4},
pages = {679-683},
publisher = {Accademia Nazionale dei Lincei},
title = {Wallman-type compaerifications and function lattices},
url = {http://eudml.org/doc/287308},
volume = {82},
year = {1988},
}

TY - JOUR
AU - Caterino, Alessandro
AU - Vipera, Maria Cristina
TI - Wallman-type compaerifications and function lattices
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1988/12//
PB - Accademia Nazionale dei Lincei
VL - 82
IS - 4
SP - 679
EP - 683
AB - Let $F \subset C^{\ast} (X)$ be a vector sublattice over $\mathbb{R}$ which separates points from closed sets of $X$. The compactification $e_{F}X$ obtained by embedding $X$ in a real cube via the diagonal map, is different, in general, from the Wallman compactification $\omega (Z(F))$. In this paper, it is shown that there exists a lattice $F_{z}$ containing $F$ such that $\omega (Z(F)) = \omega (Z(F_{z})) = e_{F}X$. In particular this implies that $\omega (Z(F)) \ge e_{F}X$. Conditions in order to be $\omega (Z(F)) = e_{F}X$ are given. Finally we prove that, if $\alpha X$ is a compactification of $X$ such that $Cl_{\alpha X} (\alpha X \setminus X)$ is $0$-dimensional, then there is an algebra $A \subset C^{ast} (X)$ such that $\omega (Z(A)) = e_{A} X = \alpha X$.
LA - eng
KW - Compactifications; Normal bases; Function lattices; Zero-sets; normal bases; function lattices; zero-sets; Wallman compactification
UR - http://eudml.org/doc/287308
ER -