Wallman-type compaerifications and function lattices

Alessandro Caterino; Maria Cristina Vipera

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1988)

  • Volume: 82, Issue: 4, page 679-683
  • ISSN: 1120-6330

Abstract

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Let F C ( X ) be a vector sublattice over which separates points from closed sets of X . The compactification e F X obtained by embedding X in a real cube via the diagonal map, is different, in general, from the Wallman compactification ω ( Z ( F ) ) . In this paper, it is shown that there exists a lattice F z containing F such that ω ( Z ( F ) ) = ω ( Z ( F z ) ) = e F X . In particular this implies that ω ( Z ( F ) ) e F X . Conditions in order to be ω ( Z ( F ) ) = e F X are given. Finally we prove that, if α X is a compactification of X such that C l α X ( α X X ) is 0 -dimensional, then there is an algebra A C a s t ( X ) such that ω ( Z ( A ) ) = e A X = α X .

How to cite

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Caterino, Alessandro, and Vipera, Maria Cristina. "Wallman-type compaerifications and function lattices." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 82.4 (1988): 679-683. <http://eudml.org/doc/287308>.

@article{Caterino1988,
abstract = {Let $F \subset C^\{\ast\} (X)$ be a vector sublattice over $\mathbb\{R\}$ which separates points from closed sets of $X$. The compactification $e_\{F\}X$ obtained by embedding $X$ in a real cube via the diagonal map, is different, in general, from the Wallman compactification $\omega (Z(F))$. In this paper, it is shown that there exists a lattice $F_\{z\}$ containing $F$ such that $\omega (Z(F)) = \omega (Z(F_\{z\})) = e_\{F\}X$. In particular this implies that $\omega (Z(F)) \ge e_\{F\}X$. Conditions in order to be $\omega (Z(F)) = e_\{F\}X$ are given. Finally we prove that, if $\alpha X$ is a compactification of $X$ such that $Cl_\{\alpha X\} (\alpha X \setminus X)$ is $0$-dimensional, then there is an algebra $A \subset C^\{ast\} (X)$ such that $\omega (Z(A)) = e_\{A\} X = \alpha X$.},
author = {Caterino, Alessandro, Vipera, Maria Cristina},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Compactifications; Normal bases; Function lattices; Zero-sets; normal bases; function lattices; zero-sets; Wallman compactification},
language = {eng},
month = {12},
number = {4},
pages = {679-683},
publisher = {Accademia Nazionale dei Lincei},
title = {Wallman-type compaerifications and function lattices},
url = {http://eudml.org/doc/287308},
volume = {82},
year = {1988},
}

TY - JOUR
AU - Caterino, Alessandro
AU - Vipera, Maria Cristina
TI - Wallman-type compaerifications and function lattices
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1988/12//
PB - Accademia Nazionale dei Lincei
VL - 82
IS - 4
SP - 679
EP - 683
AB - Let $F \subset C^{\ast} (X)$ be a vector sublattice over $\mathbb{R}$ which separates points from closed sets of $X$. The compactification $e_{F}X$ obtained by embedding $X$ in a real cube via the diagonal map, is different, in general, from the Wallman compactification $\omega (Z(F))$. In this paper, it is shown that there exists a lattice $F_{z}$ containing $F$ such that $\omega (Z(F)) = \omega (Z(F_{z})) = e_{F}X$. In particular this implies that $\omega (Z(F)) \ge e_{F}X$. Conditions in order to be $\omega (Z(F)) = e_{F}X$ are given. Finally we prove that, if $\alpha X$ is a compactification of $X$ such that $Cl_{\alpha X} (\alpha X \setminus X)$ is $0$-dimensional, then there is an algebra $A \subset C^{ast} (X)$ such that $\omega (Z(A)) = e_{A} X = \alpha X$.
LA - eng
KW - Compactifications; Normal bases; Function lattices; Zero-sets; normal bases; function lattices; zero-sets; Wallman compactification
UR - http://eudml.org/doc/287308
ER -

References

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  1. ALÓ, R.A. and SHAPIRO, H.L. (1968) - A note on compactifications and semi-normal spaces, J. Austr. Math. Soc., 8, 102-108. Zbl0161.42202MR227943
  2. BAAYEN, P.C. and VAN MILL, J. (1978) - Compactifications of locally compact spaces with zero-dimensional remainder, Top. Appl., 9, 125-129. Zbl0402.54020MR493967
  3. BALL, B.J. and YOKURA, S. (1983) - Compactifications determined by subsets of C ( X ) , II, Top. Appl., 15, 1-6. Zbl0489.54019MR676960DOI10.1016/0166-8641(83)90041-X
  4. BANDT, C. (1977) - On Wallman-Shanin compactifications, Mat. Nachr., 77, 333-351. Zbl0276.54016MR451202
  5. BILES, C.M. (1970) - Wallman-type compactifications, Proc. Am. Math. Soc., 25, 363-368. Zbl0194.54701MR263029
  6. BLASCO, J.L. (1983) - Hausdorff compactifications and Lebesgue sets; Top. Appl., 15, 111-117. Zbl0498.54021MR686089DOI10.1016/0166-8641(83)90030-5
  7. BROOKS, R.M. (1967) - On Wallman compactifications, Fund. Math., 60, 157-173. Zbl0147.41402MR210069
  8. CATERINO, A. and VIPERA, M.C. - Weight of a compactification and generating sets of functions, Rendic. Sem. Mat. Univ. Padova, to appear. Zbl0657.54020MR964018
  9. CHANDLER, R.E. (1976) - Hausdorff Compactification, Marcel Dekker, New York. Zbl0338.54001MR515002
  10. FRINK, O. (1964) - Compactifications and semi-normal spaces, Amer. J. Math., 86, 602-607. Zbl0129.38101MR166755
  11. STEINER, E.F. (1968) - Wallman spaces and compactifications, Fund. Math., 61, 295-304. Zbl0164.23501MR222849

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