Rings of continuous functions vanishing at infinity

Aliabad, Ali Rezaei, Azarpanah, F., and Namdari, M.. "Rings of continuous functions vanishing at infinity." Commentationes Mathematicae Universitatis Carolinae 45.3 (2004): 519-533. <http://eudml.org/doc/249355>.

@article{Aliabad2004,
abstract = {We prove that a Hausdorff space $X$ is locally compact if and only if its topology coincides with the weak topology induced by $C_\infty (X)$. It is shown that for a Hausdorff space $X$, there exists a locally compact Hausdorff space $Y$ such that $C_\infty (X)\cong C_\infty (Y)$. It is also shown that for locally compact spaces $X$ and $Y$, $C_\infty (X)\cong C_\infty (Y)$ if and only if $X\cong Y$. Prime ideals in $C_\infty (X)$ are uniquely represented by a class of prime ideals in $C^*(X)$. $\infty $-compact spaces are introduced and it turns out that a locally compact space $X$ is $\infty $-compact if and only if every prime ideal in $C_\infty (X)$ is fixed. The existence of the smallest $\infty $-compact space in $\beta X$ containing a given space $X$ is proved. Finally some relations between topological properties of the space $X$ and algebraic properties of the ring $C_\infty (X)$ are investigated. For example we have shown that $C_\infty (X)$ is a regular ring if and only if $X$ is an $\infty $-compact $\operatorname\{P\}_\infty $-space.},
author = {Aliabad, Ali Rezaei, Azarpanah, F., Namdari, M.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$\sigma $-compact; pseudocompact; $\infty $-compact; $\infty $-compactification; $\operatorname\{P\}_\{\infty \}$-space; P-point; regular ring; fixed and free ideals; function vanishing at infinity; functions with compact support; locally compact space; ideal; regular ring},
language = {eng},
number = {3},
pages = {519-533},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Rings of continuous functions vanishing at infinity},
url = {http://eudml.org/doc/249355},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Aliabad, Ali Rezaei
AU - Azarpanah, F.
AU - Namdari, M.
TI - Rings of continuous functions vanishing at infinity
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 3
SP - 519
EP - 533
AB - We prove that a Hausdorff space $X$ is locally compact if and only if its topology coincides with the weak topology induced by $C_\infty (X)$. It is shown that for a Hausdorff space $X$, there exists a locally compact Hausdorff space $Y$ such that $C_\infty (X)\cong C_\infty (Y)$. It is also shown that for locally compact spaces $X$ and $Y$, $C_\infty (X)\cong C_\infty (Y)$ if and only if $X\cong Y$. Prime ideals in $C_\infty (X)$ are uniquely represented by a class of prime ideals in $C^*(X)$. $\infty $-compact spaces are introduced and it turns out that a locally compact space $X$ is $\infty $-compact if and only if every prime ideal in $C_\infty (X)$ is fixed. The existence of the smallest $\infty $-compact space in $\beta X$ containing a given space $X$ is proved. Finally some relations between topological properties of the space $X$ and algebraic properties of the ring $C_\infty (X)$ are investigated. For example we have shown that $C_\infty (X)$ is a regular ring if and only if $X$ is an $\infty $-compact $\operatorname{P}_\infty $-space.
LA - eng
KW - $\sigma $-compact; pseudocompact; $\infty $-compact; $\infty $-compactification; $\operatorname{P}_{\infty }$-space; P-point; regular ring; fixed and free ideals; function vanishing at infinity; functions with compact support; locally compact space; ideal; regular ring
UR - http://eudml.org/doc/249355
ER -