### Impacto

Garrido Carballo, M. Isabel and Jaramillo Aguado, Jesús Ángel
(2004)
*Homomorphisms on function lattices.*
Monatshefte fur Mathematik, 141
.
pp. 127-146.
ISSN 0026-9255

PDF
Restringido a Repository staff only hasta 2020. 308kB |

Official URL: http://www.springerlink.com/content/5w7gyp740qxky1nh/fulltext.pdf

## Abstract

In this paper we study real lattice homomorphisms on a unital vector lattice L subset of C(X), where X is a completely regular space. We stress on topological properties of its structure spaces and on its representation as point evaluations. These results are applied to the lattice L = Lip(X) of real Lipschitz functions on a metric space. Using the automatic continuity of lattice homomorphisms with respect to the Lipschitz norm, we are able to derive a Banach-Stone theorem in this context. Namely, it is proved that the unital vector lattice structure of Lip (X) characterizes the Lipschitz structure of the complete metric space X. In the case L = Lip (X) of bounded Lipschitz functions, an analogous result is obtained in the class of complete quasiconvex metric spaces.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Lattices of continuous functions, lattice homomorphisms, Lipschitz functions, Banach-Stone-theorems |

Subjects: | Sciences > Mathematics > Functional analysis and Operator theory Sciences > Mathematics > Topology |

ID Code: | 15519 |

References: | Araujo J, Beckenstein E, Narici L (1995) Biseparating maps and homeomorphic realcompactifications.J Math Anal Appl 192: 258–265 Araujo J, Font JJ (1997) Linear isometries between subspaces of continuous functions. Trans Amer Math Soc 349: 413–428 Arrazola JR, Bustamante J (2000) Homomorphisms on Lipschitz spaces. Monatsh Math 129: 25–30 Bessaga C (1966) Every infinite-dimensional Hilbert space is diffeomorphic with its unit sphere.Bull Acad Polon Sci 14: 27–31 Bridson MR, Haefliger A (1999) Metric Spaces of Non-Positive Curvature. Berlin: Springer Efremovich VA (1952) The geometry of proximity I. Math Sbor 31: 189–200 Engelking R (1977) General Topology. Warsaw: PWN-Polish Scientific Garrido MI, Gomez J, Jaramillo JA (1994) Homomorphisms on some function algebras. Can J Math 46: 734–745 Garrido MI, Jaramillo JA (2000) A Banach-Stone theorem for uniformly continuous functions.Monatsh Math 131: 189–192 Garrido MI, Jaramillo JA (2001) Representation of homomorphisms on function lattices. Rend Istit Math Univ Trieste 32: 73–79 Garrido MI, Montalvo F (1992) Uniform approximation theorems for real-valued functions. Topol Appl 45: 145–155 Gelfand I, Kolmogorof A (1939) On rings of continuous functions on topological spaces. Dokl Akad Nauk 22: 11–15 Gillman L, Jerison J (1976) Rings of Continuous Functions. New-York: Springer Herna´ndez S (1999) Uniformly continuous mappings defined by isometries of spaces of bounded uniformly continuous functions. Topology Atlas No. 394 (http:==at.yorku.ca=i=d=e=b=18.htm) Hewitt E (1948) Rings of real-valued continuous functions, I. Trans Amer Math Soc 64: 54–99 Isbell JR (1958) Algebras of uniformly continuous functions. Ann Math 68: 96–125 Izzo AJ (1994) Locally uniformly continuous functions. Proc Amer Math Soc 122: 1095–1100 Jech T (1978) Set Theory. New York Academic Press Kaplansky IL (1947) Topological rings. Amer J Math Appl 69: 153–183 Kriegl A, Michor P (1997) The Convenient Setting of Global Analysis. Providence, RI: Amer Math Soc Lacruz M, Llavona JG (1997) Composition operators between algebras of uniformly continuous functions. Archiv Math 69: 52–56 McShane EJ (1934) Extension of range of functions. Bull Amer Math Soc 40: 837–842 Nachbin L (1967) Elements of Approximation Theory. Princeton, NJ: Van Nostrand Ransford TJ (1995) Characters and point evaluations. Canad Math Bull 38: 237–241 Redlin L,Watson S (1997) Structure spaces for rings of continuous functions with applications to realcompactifications. Fund Math 152: 151–163 Scanlon, CH (1970) Rings of functions with certain Lipschitz properties. Pacific J Math 32: 197–201 Sherbert DR (1963) Banach algebras of Lipschitz functions. Pacific J Math 13: 1387–1399 Shirota T (1952) A generalization of a theorem of I. Kaplanski. Osaka Math J 4: 121–132 Su, Li Pi (1968) Algebraic properties of certain rings of continuous functions. Pacific J Math 27: 175–191 Weaver N (1994) Lattices of Lipschitz functions. Pacific J Math 164: 179–193 Weaver N (1995) Order completeness in Lipschitz algebras. J Funct Anal 130: 118–130 Weaver N (1999) Lipschitz Algebras. Singapore: World Scientific Woods RG (1995) The minimum uniform compactification of a metric space. Fund Math 147: 39–59 |

Deposited On: | 07 Jun 2012 08:25 |

Last Modified: | 06 Feb 2014 10:26 |

Repository Staff Only: item control page