Automatic continuity of biseparating homomorphisms defined between groups of continuous functions

Ferrer, María V.; Hernández, Salvador; Ródenas, Ana María Bofill

doi:10.1016/j.topol.2009.04.069

Cited by 3 publications

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“…There are many precedents in the study the representation of group homomorphisms for group-valued continuous functions. Among them, the following ones are relevant here (cf [3,5,9,16,8,18,20,21,24,25]). Most basic facts and notions related to topological properties may be found in [12,6].…”

Section: Introductionmentioning

confidence: 99%

Representation of group isomorphisms I

Ferrer

Gary

Hernández

2019

Topology and its Applications

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Let G be a metric group and let Aut(G) denote the automorphism group of G. If A and B are groups of G-valued maps defined on the sets X and Y , respectively, we say that A and B are equivalent if there is a group isomorphism H :In this case, we say that H is represented as a weighted composition operator. A group isomorphism H defined between A and B is called separating when for each pair of maps f, g ∈ A satisfying that f −1 (e G ) ∪ g −1 (e G ) = X, it holds that (Hf ) −1 (e G ) ∪ (Hg) −1 (e G ) = Y . Our main result establishes that under some mild conditions, every separating group isomorphism can be represented as a weighted composition operator. As a consequence we establish the equivalence of two function groups if there is a biseparating isomorphism defined between them.

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Section: Introductionmentioning

confidence: 99%

Representation of group isomorphisms I

Ferrer

Gary

Hernández

2019

Topology and its Applications

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“…There are many precedents in the study the representation of group homomorphisms for group-valued continuous functions. Among them, the following ones are relevant here (cf [3,6,16,20,21,23,28,29]). Most basic facts and notions related to topological properties may be found in [5].…”

Section: Introductionmentioning

confidence: 99%

Representation of Group Isomorphisms: The Compact Case

Ferrer

Gary

Hernández

2015

Journal of Function Spaces

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Let G be a discrete group and let A and B be two subgroups of Gvalued continuous functions defined on two 0-dimensional compact spaces X and Y . A group isomorphism H defined between A and B is called separating when for each pair of maps f, g ∈ A satisfying that f −1 (e G ) ∪ g −1 (e G ) = X, it holds that Hf −1 (e G )∪Hg −1 (e G ) = Y . We prove that under some mild conditions every separating isomorphism H : A −→ B can be represented by means of a continuous function h : Y −→ X as a weighted composition operator. As a consequence we establish the equivalence of two subgroups of continuous functions if there is a biseparating isomorphism defined between them.

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“…Existen muchos precedentes en el estudio de la representación de homomorfismos de grupos para funciones continuas que toman valores en un grupo. Se pueden consultar, como se dijo en el segundo capítulo, en las siguientes referencias (ver [5], [9], [20], [24], [26], [29], [36], [37]). La mayoría de los hechos básicos y las nociones relacionadas con propiedades topológicas se pueden encontrar en [6].…”

Section: Introductionunclassified

Representación de isomorfismos entre espacios de funciones continuas

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grupos topológicos y U un operador definido en E y cuyo codominio es E 1 . Decimos que U es lineal si es aditivo y continuo.A continuación se mencionarán algunas propiedades de los operadores lineales definidos entre espacios vectoriales normados, no necesariamente completos. 1.1.1. Propiedades de operadores lineales. Teorema 1.1.5 (Cap. IV, teorema 1 de [1]). Sean E, G y E 1 espacios vectoriales normados tales que G ⊆ E. Si U : G → E 1 es un operador aditivo, entonces U es lineal si, y sólo si, existe un número M tal que U (x) ≤ M x , para todo x ∈ G. Sean E y E 1 espacios vectoriales normados. Para un operador lineal U : G → E 1 , definido sobre un espacio vectorial G ⊆ E, la norma del operador U en G, denotada por U G , es el número real más pequeño, M , que satisface la siguiente condición: U (x) ≤ M x , para todo x ∈ G.

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Automatic continuity of biseparating homomorphisms defined between groups of continuous functions

Cited by 3 publications

References 14 publications

Representation of group isomorphisms I

Representation of group isomorphisms I

Representation of Group Isomorphisms: The Compact Case

Representación de isomorfismos entre espacios de funciones continuas

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