Rings of continuous functions vanishing at infinity

Ali Rezaei Aliabad; F. Azarpanah; M. Namdari

Commentationes Mathematicae Universitatis Carolinae (2004)

  • Volume: 45, Issue: 3, page 519-533
  • ISSN: 0010-2628

Abstract

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We prove that a Hausdorff space X is locally compact if and only if its topology coincides with the weak topology induced by C ( X ) . It is shown that for a Hausdorff space X , there exists a locally compact Hausdorff space Y such that C ( X ) C ( Y ) . It is also shown that for locally compact spaces X and Y , C ( X ) C ( Y ) if and only if X Y . Prime ideals in C ( X ) are uniquely represented by a class of prime ideals in C * ( X ) . -compact spaces are introduced and it turns out that a locally compact space X is -compact if and only if every prime ideal in C ( X ) is fixed. The existence of the smallest -compact space in β 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 ( X ) are investigated. For example we have shown that C ( X ) is a regular ring if and only if X is an -compact P -space.

How to cite

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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 -

References

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  4. Azarpanah F., Sondararajan T., When the family of functions vanishing at infinity is an ideal of C ( X ) , Rocky Mountain J. Math. 31.4 (2001), 1-8. (2001) MR1895289
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  7. Gillman L., Jerison M., Rings of Continuous Functions, Springer, New York, 1976. Zbl0327.46040MR0407579
  8. Goodearl K.R., Warfield R.B., Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge Univ. Press, Cambridge, 1989. Zbl1101.16001MR1020298
  9. Kohls C.W., Ideals in rings of continuous functions, Fund. Math. 45 (1957), 28-50. (1957) Zbl0079.32701MR0102731
  10. McConnel J.C., Robson J.C., Noncommutative Noetherian Rings, Wiley Interscience, New York, 1987. MR0934572
  11. Namdari M., Algebraic properties of C ( X ) , Proceeding of Abstracts of Short Communications and Poster Sessions, ICM 2002, p.85. 
  12. Rudd D., On isomorphisms between ideals in rings of continuous functions, Trans. Amer. Math. Soc. 159 (1971), 335-353. (1971) Zbl0228.46019MR0283575

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