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Interpolation sets and the size of quotients of function spaces on a locally compact group


Authors: Mahmoud Filali and Jorge Galindo
Journal: Trans. Amer. Math. Soc. 369 (2017), 575-603
MSC (2010): Primary 22D15; Secondary 43A46, 43A15, 43A60
DOI: https://doi.org/10.1090/tran6662
Published electronically: March 9, 2016
MathSciNet review: 3557786
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Abstract: We devise a fairly general method for estimating the size of quotients between algebras of functions on a locally compact group. This method is based on the concept of interpolation set we introduced and studied recently and unifies the approaches followed by many authors to obtain particular cases.

We find in this way that there is a linear isometric copy of $ \ell _\infty (\kappa )$ in each of the following quotient spaces:

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$ \mathscr {WAP}_0(G)/C_0(G)$ whenever $ G$ contains a subset $ X$ that is an $ E$-set (see the definition in the paper) and $ \kappa =\kappa (X)$ is the minimal number of compact sets required to cover $ X$. In particular, $ \kappa =\kappa (G)$ when $ G$ is an $ SIN$-group.
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$ \mathscr {WAP}(G)/\mathscr {B}(G)$, when $ G$ is any locally compact group and $ \kappa =\kappa (Z(G))$ and $ Z(G)$ is the centre of $ G$, or when $ G$ is either an $ IN$-group or a nilpotent group and $ \kappa =\kappa (G)$.
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$ \mathscr {WAP}_0(G)/\mathscr {B}_0(G)$, when $ G$ and $ \kappa $ are as in the foregoing item.
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$ \mathscr {CB}(G)/\mathscr {LUC}(G)$, when $ G$ is any locally compact group that is neither compact nor discrete and $ \kappa =\kappa (G)$.

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Additional Information

Mahmoud Filali
Affiliation: Department of Mathematical Sciences, University of Oulu, Oulu, Finland
Email: mfilali@cc.oulu.fi

Jorge Galindo
Affiliation: Instituto Universitario de Matemáticas y Aplicaciones (IMAC), Universidad Jaume I, E-12071, Castellón, Spain
Email: jgalindo@mat.uji.es

DOI: https://doi.org/10.1090/tran6662
Keywords: Almost periodic functions, Fourier-Stieltjes algebra, weakly almost periodic, semigroup compactification, almost periodic compactification, interpolation sets
Received by editor(s): March 27, 2014
Received by editor(s) in revised form: November 5, 2014, and January 10, 2015
Published electronically: March 9, 2016
Additional Notes: The research of the second author was partially supported by the Spanish Ministry of Science (including FEDER funds), grant MTM2011-23118 and Fundació Caixa Castelló-Bancaixa, grant number P1$⋅$1B2014-35.
Article copyright: © Copyright 2016 American Mathematical Society