| Title:
|
Rings of continuous functions vanishing at infinity (English) |
| Author:
|
Aliabad, A. R. |
| Author:
|
Azarpanah, F. |
| Author:
|
Namdari, M. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
45 |
| Issue:
|
3 |
| Year:
|
2004 |
| Pages:
|
519-533 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
$\sigma $-compact |
| Keyword:
|
pseudocompact |
| Keyword:
|
$\infty $-compact |
| Keyword:
|
$\infty $-compactification |
| Keyword:
|
$\operatorname{P}_{\infty }$-space |
| Keyword:
|
P-point |
| Keyword:
|
regular ring |
| Keyword:
|
fixed and free ideals |
| MSC:
|
54C40 |
| MSC:
|
54D45 |
| idZBL:
|
Zbl 1097.54021 |
| idMR:
|
MR2103146 |
| . |
| Date available:
|
2009-05-05T16:47:06Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119479 |
| . |
| Reference:
|
[1] Al-ezeh H., Natsheh M.A., Hussein D.: Some properties of the ring of continuous functions.Arch. Math. 51 (1988), 51-60. Zbl 0638.54017, MR 0954069 |
| Reference:
|
[2] Azarpanah F.: Essential ideals in $C(X)$.Period. Math. Hungar. 31 2 (1995), 105-112. Zbl 0869.54021, MR 1609417 |
| Reference:
|
[3] Azarpanah F., Karamzadeh O.A.S.: Algebraic characterizations of some disconnected spaces.Ital. J. Pure Appl. Math., no. 12 (2002), 155-168. Zbl 1117.54030, MR 1962109 |
| Reference:
|
[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. MR 1895289 |
| Reference:
|
[5] Berberian S.K.: Baer*-rings.Springer, New York-Berlin, 1972. Zbl 0534.16011, MR 0429975 |
| Reference:
|
[6] Engelking R.: General Topology.PWN-Polish Scientific Publishing, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[7] Gillman L., Jerison M.: Rings of Continuous Functions.Springer, New York, 1976. Zbl 0327.46040, MR 0407579 |
| Reference:
|
[8] Goodearl K.R., Warfield R.B., Jr.: An Introduction to Noncommutative Noetherian Rings.Cambridge Univ. Press, Cambridge, 1989. Zbl 1101.16001, MR 1020298 |
| Reference:
|
[9] Kohls C.W.: Ideals in rings of continuous functions.Fund. Math. 45 (1957), 28-50. Zbl 0079.32701, MR 0102731 |
| Reference:
|
[10] McConnel J.C., Robson J.C.: Noncommutative Noetherian Rings.Wiley Interscience, New York, 1987. MR 0934572 |
| Reference:
|
[11] Namdari M.: Algebraic properties of $C_\infty(X)$.Proceeding of Abstracts of Short Communications and Poster Sessions, ICM 2002, p.85. |
| Reference:
|
[12] Rudd D.: On isomorphisms between ideals in rings of continuous functions.Trans. Amer. Math. Soc. 159 (1971), 335-353. Zbl 0228.46019, MR 0283575 |
| . |
