Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
$\sigma $-compact; pseudocompact; $\infty $-compact; $\infty $-compactification; $\operatorname{P}_{\infty }$-space; P-point; regular ring; fixed and free ideals
$\sigma $-compact; pseudocompact; $\infty $-compact; $\infty $-compactification; $\operatorname{P}_{\infty }$-space; P-point; regular ring; fixed and free ideals
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.
References:
[1] Al-ezeh H., Natsheh M.A., Hussein D.: Some properties of the ring of continuous functions. Arch. Math. 51 (1988), 51-60. MR 0954069 | Zbl 0638.54017
[2] Azarpanah F.: Essential ideals in $C(X)$. Period. Math. Hungar. 31 2 (1995), 105-112. MR 1609417 | Zbl 0869.54021
[3] Azarpanah F., Karamzadeh O.A.S.: Algebraic characterizations of some disconnected spaces. Ital. J. Pure Appl. Math., no. 12 (2002), 155-168. MR 1962109 | Zbl 1117.54030
[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
[5] Berberian S.K.: Baer*-rings. Springer, New York-Berlin, 1972. MR 0429975 | Zbl 0534.16011
[6] Engelking R.: General Topology. PWN-Polish Scientific Publishing, 1977. MR 0500780 | Zbl 0684.54001
[7] Gillman L., Jerison M.: Rings of Continuous Functions. Springer, New York, 1976. MR 0407579 | Zbl 0327.46040
[8] Goodearl K.R., Warfield R.B., Jr.: An Introduction to Noncommutative Noetherian Rings. Cambridge Univ. Press, Cambridge, 1989. MR 1020298 | Zbl 1101.16001
[9] Kohls C.W.: Ideals in rings of continuous functions. Fund. Math. 45 (1957), 28-50. MR 0102731 | Zbl 0079.32701
[10] McConnel J.C., Robson J.C.: Noncommutative Noetherian Rings. Wiley Interscience, New York, 1987. MR 0934572
[11] Namdari M.: Algebraic properties of $C_\infty(X)$. 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. MR 0283575 | Zbl 0228.46019
Search
Partner of
