Correspondences between ideals and $z$-filters for rings of continuous functions between $C^{\ast }$ and $C$

Phyllis Panman, Joshua Sack, and Saleem Watson. "Correspondences between ideals and $z$-filters for rings of continuous functions between $C^∗$ and $C$." Commentationes Mathematicae 52.1 (2012): null. <http://eudml.org/doc/291864>.

@article{PhyllisPanman2012,
abstract = {Let $X$ be a completely regular topological space. Let $A(X)$ be a ring of continuous functions between $C^∗(X)$ and $C(X)$, that is, $C^∗(X) \subset A(X) \subset C(X)$. In [9], a correspondence $\mathcal \{Z\}_A$ between ideals of $A(X)$ and $z$-filters on $X$ is defined. Here we show that $\mathcal \{Z\}_A$ extends the well-known correspondence for $C^∗(X)$ to all rings $A(X)$. We define a new correspondence $\mathcal \{Z\}_A$ and show that it extends the well-known correspondence for $C(X)$ to all rings $A(X)$. We give a formula that relates the two correspondences. We use properties of $\mathcal \{Z\}_A$ and $\mathcal \{Z\}_A$ to characterize $C^∗(X)$ and $C(X)$ among all rings $A(X)$. We show that $\mathcal \{Z\}_A$ defines a one-one correspondence between maximal ideals in $A(X)$ and the $z$-ultrafilters in $X$.},
author = {Phyllis Panman, Joshua Sack, Saleem Watson},
journal = {Commentationes Mathematicae},
keywords = {Rings of continuous functions; Ideals; $z$-filters; Kernel; Hull},
language = {eng},
number = {1},
pages = {null},
title = {Correspondences between ideals and $z$-filters for rings of continuous functions between $C^∗$ and $C$},
url = {http://eudml.org/doc/291864},
volume = {52},
year = {2012},
}

TY - JOUR
AU - Phyllis Panman
AU - Joshua Sack
AU - Saleem Watson
TI - Correspondences between ideals and $z$-filters for rings of continuous functions between $C^∗$ and $C$
JO - Commentationes Mathematicae
PY - 2012
VL - 52
IS - 1
SP - null
AB - Let $X$ be a completely regular topological space. Let $A(X)$ be a ring of continuous functions between $C^∗(X)$ and $C(X)$, that is, $C^∗(X) \subset A(X) \subset C(X)$. In [9], a correspondence $\mathcal {Z}_A$ between ideals of $A(X)$ and $z$-filters on $X$ is defined. Here we show that $\mathcal {Z}_A$ extends the well-known correspondence for $C^∗(X)$ to all rings $A(X)$. We define a new correspondence $\mathcal {Z}_A$ and show that it extends the well-known correspondence for $C(X)$ to all rings $A(X)$. We give a formula that relates the two correspondences. We use properties of $\mathcal {Z}_A$ and $\mathcal {Z}_A$ to characterize $C^∗(X)$ and $C(X)$ among all rings $A(X)$. We show that $\mathcal {Z}_A$ defines a one-one correspondence between maximal ideals in $A(X)$ and the $z$-ultrafilters in $X$.
LA - eng
KW - Rings of continuous functions; Ideals; $z$-filters; Kernel; Hull
UR - http://eudml.org/doc/291864
ER -