Remarks on $LBI$-subalgebras of $C\left(X\right)$

@article{Parsinia2016,
abstract = {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.},
author = {Parsinia, Mehdi},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {local bounded inversion; structure space; $z_A^\beta $-ideal; complete ring of functions},
language = {eng},
number = {2},
pages = {261-270},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Remarks on $LBI$-subalgebras of $C(X)$},
url = {http://eudml.org/doc/280135},
volume = {57},
year = {2016},
}

TY - JOUR
AU - Parsinia, Mehdi
TI - Remarks on $LBI$-subalgebras of $C(X)$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2016
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 57
IS - 2
SP - 261
EP - 270
AB - 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.
LA - eng
KW - local bounded inversion; structure space; $z_A^\beta $-ideal; complete ring of functions
UR - http://eudml.org/doc/280135
ER -