arXiv subjects

Marita Ferrer

Publications and source records attributed to Marita Ferrer.

4 recordsLinked to original sources

arXiv2018

Representation of Group Isomorphisms I

Let $G$ be a metric group and let $\sA ut(G)$ denote the automorphism group of $G$. If $\sA$ and $\sB$ are groups of $G$-valued maps defined on the sets $X$ and $Y$, respectively, we say that $\sA$ and $\sB$ are \emph{equivalent} if there is a group isomorphism $H\colon\sA\to\sB$ such that there is a bijective map $h\colon Y\to X$ and a map $w\colon Y\to \sA ut (G)$ satisfying $Hf(y)=w[y](f(h(y)))$ for all $y\in Y$ and $f\in \sA$. In this case, we say that $H$ is represented as a \emph{weighted composition operator}. A group isomorphism $H$ defined between $\sA$ and $\sB$ is called \emph{separating} when for each pair of maps $f,g\in \sA$ satisfying that $f^{-1}(e_G)\cup g^{-1}(e_G)=X$, it holds that $(Hf)^{-1}(e_G)\cup (Hg)^{-1}(e_G)=Y$. 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.

math.GN↗

arXiv2018

A dichotomy property for locally compact groups

We extend to metrizable locally compact groups Rosenthal's theorem describing those Banach spaces containing no copy of $l_1$. For that purpose, we transfer to general locally compact groups the notion of interpolation ($I_0$) set, which was defined by Hartman and Ryll-Nardzewsky [25] for locally compact abelian groups. Thus we prove that for every sequence $\lbrace g_n \rbrace_{n<ω}$ in a locally compact group $G$, then either $\lbrace g_n \rbrace_{n<ω}$ has a weak Cauchy subsequence or contains a subsequence that is an $I_0$ set. This result is subsequently applied to obtain sufficient conditions for the existence of Sidon sets in a locally compact group $G$, an old question that remains open since 1974 (see [32] and [20]). Finally, we show that every locally compact group strongly respects compactness extending thereby a result by Comfort, Trigos-Arrieta, and Wu [13], who established this property for abelian locally compact groups.

math.GN↗

arXiv2016

Equicontinuity criteria for metric-valued sets of continuous functions

Combining ideas of Troallic and Cascales, Namioka, and Vera, we prove several characterizations of \textit{almost equicontinuity} and \textit{hereditary almost equicontinuity} for subsets of metric-valued continuous functions when they are defined on a Čech-complete space. We also obtain some applications of these results to topological groups and dynamical systems.

math.GN↗

arXiv2015

Weight-preserving isomorphisms between spaces of continuous functions: The scalar case

Let $\mathbb F$ be a finite field and let $\mathcal A$ and $\mathcal B$ be vector spaces of $\mathbb F$-valued continuous functions defined on locally compact spaces $X$ and $Y$, respectively. We look at the representation of linear bijections $H:\mathcal A\longrightarrow \mathcal B$ by continuous functions $h:Y\longrightarrow X$ as weighted composition operators. In order to do it, we extend the notion of Hamming metric to infinite spaces. Our main result establishes that under some mild conditions, every Hamming isometry can be represented as a weighted composition operator. Connections to coding theory are also highlighted.

math.FA↗