| Title:
|
Remarks on $LBI$-subalgebras of $C(X)$ (English) |
| Author:
|
Parsinia, Mehdi |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
57 |
| Issue:
|
2 |
| Year:
|
2016 |
| Pages:
|
261-270 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $A(X)$ denote a subalgebra of $C(X)$ which is closed under local bounded inversion, briefly, an $LBI$-subalgebra. These subalgebras were first introduced and studied in Redlin L., Watson S., Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151--163. By characterizing maximal ideals of $A(X)$, we generalize the notion of $z_A^\beta$-ideals, which was first introduced in Acharyya S.K., De D., An interesting class of ideals in subalgebras of $C(X)$ containing $C^*(X)$, Comment. Math. Univ. Carolin. 48 (2007), 273--280 for intermediate subalgebras, to the $LBI$-subalgebras. Using these, it is simply shown that the structure space of every $LBI$-subalgebra is homeomorphic with a quotient of $\beta X$. This gives a different approach to the results of Redlin L., Watson S., Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151--163 and also shows that the Banaschewski-compactification of a zero-dimensional space $X$ is a quotient of $\beta X$. Finally, we consider the class of complete rings of functions which was first defined in Byun H.L., Redlin L., Watson S., Local invertibility in subrings of $C^*(X)$, Bull. Austral. Math. Soc. 46 (1992), 449--458. Showing that every such subring is an $LBI$-subalgebra, we prove that the compactification of $X$ associated to each complete ring of functions, which is identified in Byun H.L., Redlin L., Watson S., Local invertibility in subrings of $C^*(X)$, Bull. Austral. Math. Soc. 46 (1992), 449--458 via the mapping ${\mathcal Z}_A$, is in fact, the structure space of that subring. Henceforth, some statements in Byun H.L., Redlin L., Watson S., Local invertibility in subrings of $C^*(X)$, Bull. Austral. Math. Soc. 46 (1992), 449--458 could be proved in a different way. (English) |
| Keyword:
|
local bounded inversion |
| Keyword:
|
structure space |
| Keyword:
|
$z_A^\beta$-ideal |
| Keyword:
|
complete ring of functions |
| MSC:
|
46E25 |
| MSC:
|
54C30 |
| idZBL:
|
Zbl 1363.54031 |
| idMR:
|
MR3513449 |
| DOI:
|
10.14712/1213-7243.2015.158 |
| . |
| Date available:
|
2016-07-05T15:13:54Z |
| Last updated:
|
2018-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145750 |
| . |
| Reference:
|
[1] Acharyya S.K., De D.: $A$-compactness and minimal subalgebras of $C(X)$.Rocky Mountain J. Math. 35 (2005), no. 4, 1061–1067. MR 2178974, 10.1216/rmjm/1181069673 |
| Reference:
|
[2] Acharyya S.K., De D.: An interesting class of ideals in subalgebras of $C(X)$ containing $C^*(X)$.Comment. Math. Univ. Carolin. 48 (2007), 273–280. MR 2338095 |
| Reference:
|
[3] Bhattacharjee P., Knox M.L., McGovern W.W.: The classical ring of quotients of $C_c(X)$.Appl. Gen. Topol. 15 (2014), no. 2, 147–154. Zbl 1305.54030, MR 3267269, 10.4995/agt.2014.3181 |
| Reference:
|
[4] Byun H.L., Redlin L., Watson S.: Local bounded inversion in rings of continuous functions.Comment. Math. Univ. Carolin. 37 (1997), 37–46. Zbl 0903.54009, MR 1608229 |
| Reference:
|
[5] Byun H.L., Redlin L., Watson S.: Local invertibility in subrings of $C^*(X)$.Bull. Austral. Math. Soc. 46 (1992), 449–458. MR 1190348, 10.1017/S0004972700012119 |
| Reference:
|
[6] Byun H.L., Watson S.: Prime and maximal ideals in subrings of $C(X)$.Topology Appl. 40 (1991), 45–62. Zbl 0732.54016, MR 1114090, 10.1016/0166-8641(91)90057-S |
| Reference:
|
[7] De D., Acharyya S.K.: Characterization of function rings between $C^*(X)$ and $C(X)$.Kyungpook Math. J. 40 (2006), 503–507. MR 2282652 |
| Reference:
|
[8] Ghadermazi M., Karamzadeh O.A.S., Namdari M.: On the functionally countable subalgebra of $C(X)$.Rend. Sem. Mat. Univ. Padova 129 (2013), 47–69. Zbl 1279.54015, MR 3090630, 10.4171/RSMUP/129-4 |
| Reference:
|
[9] Gillman L., Jerison M.: Rings of Continuous Functions.Springer, New York, 1978. Zbl 0327.46040, MR 0407579 |
| Reference:
|
[10] Henriksen M., Johnson D.G.: On the struture of a class of archimedean lattice-ordered algebras.Fund. Math. 50 (1961), 73–94. MR 0133698, 10.4064/fm-50-1-73-94 |
| Reference:
|
[11] Johnson D.G., Mandelker M.: Functions with pseudocompact support.General Topology Appl. 3 (1973), 331–338. Zbl 0277.54009, MR 0331310, 10.1016/0016-660X(73)90020-2 |
| Reference:
|
[12] Koushesh M.R.: The partially ordered set of one-point extensions.Topology Appl. 158 (2011), 509–532. Zbl 1216.54007, MR 2754374, 10.1016/j.topol.2010.12.001 |
| Reference:
|
[13] Plank D.: On a class of subalgebras of $C(X)$ with applications to $\beta X-X$.Fund. Math. 64 (1969), 41–54. MR 0244953, 10.4064/fm-64-1-41-54 |
| Reference:
|
[14] Redlin H., Watson S.: Maximal ideals in subalgebras of $C(X)$.Proc. Amer. Math. Soc. 100 (1987), 763–766. Zbl 0622.54011, MR 0894451 |
| Reference:
|
[15] Redlin L., Watson S.: Structure spaces for rings of continuous functions with applications to realcompactifications.Fund. Math. 152 (1997), 151–163. Zbl 0877.54015, MR 1441231 |
| Reference:
|
[16] Rudd D.: On isomorphism between ideals in rings of continuous functions.Trans. Amer. Math. Soc. 159 (1971), 335–353. MR 0283575, 10.1090/S0002-9947-1971-0283575-1 |
| Reference:
|
[17] Rudd D.: On structure spaces of ideals in rings of continuous functions.Trans. Amer. Math. Soc. 190 (1974), 393–403. Zbl 0288.46025, MR 0350690, 10.1090/S0002-9947-1974-0350690-6 |
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