Previous |  Up |  Next

# Article

Full entry | Fulltext not available (moving wall 24 months)
Keywords:
local bounded inversion; structure space; $z_A^\beta$-ideal; complete ring of functions
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.
References:
[1] Acharyya S.K., De D.: $A$-compactness and minimal subalgebras of $C(X)$. Rocky Mountain J. Math. 35 (2005), no. 4, 1061–1067. DOI 10.1216/rmjm/1181069673 | MR 2178974
[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
[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. DOI 10.4995/agt.2014.3181 | MR 3267269 | Zbl 1305.54030
[4] Byun H.L., Redlin L., Watson S.: Local bounded inversion in rings of continuous functions. Comment. Math. Univ. Carolin. 37 (1997), 37–46. MR 1608229 | Zbl 0903.54009
[5] Byun H.L., Redlin L., Watson S.: Local invertibility in subrings of $C^*(X)$. Bull. Austral. Math. Soc. 46 (1992), 449–458. DOI 10.1017/S0004972700012119 | MR 1190348
[6] Byun H.L., Watson S.: Prime and maximal ideals in subrings of $C(X)$. Topology Appl. 40 (1991), 45–62. DOI 10.1016/0166-8641(91)90057-S | MR 1114090 | Zbl 0732.54016
[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
[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. DOI 10.4171/RSMUP/129-4 | MR 3090630 | Zbl 1279.54015
[9] Gillman L., Jerison M.: Rings of Continuous Functions. Springer, New York, 1978. MR 0407579 | Zbl 0327.46040
[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
[11] Johnson D.G., Mandelker M.: Functions with pseudocompact support. General Topology Appl. 3 (1973), 331–338. DOI 10.1016/0016-660X(73)90020-2 | MR 0331310 | Zbl 0277.54009
[12] Koushesh M.R.: The partially ordered set of one-point extensions. Topology Appl. 158 (2011), 509–532. DOI 10.1016/j.topol.2010.12.001 | MR 2754374 | Zbl 1216.54007
[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
[14] Redlin H., Watson S.: Maximal ideals in subalgebras of $C(X)$. Proc. Amer. Math. Soc. 100 (1987), 763–766. MR 0894451 | Zbl 0622.54011
[15] Redlin L., Watson S.: Structure spaces for rings of continuous functions with applications to realcompactifications. Fund. Math. 152 (1997), 151–163. MR 1441231 | Zbl 0877.54015
[16] Rudd D.: On isomorphism between ideals in rings of continuous functions. Trans. Amer. Math. Soc. 159 (1971), 335–353. DOI 10.1090/S0002-9947-1971-0283575-1 | MR 0283575
[17] Rudd D.: On structure spaces of ideals in rings of continuous functions. Trans. Amer. Math. Soc. 190 (1974), 393–403. DOI 10.1090/S0002-9947-1974-0350690-6 | MR 0350690 | Zbl 0288.46025

Partner of