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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Interpolation sets and the size of quotients of function spaces on a locally compact group
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by Mahmoud Filali and Jorge Galindo PDF
Trans. Amer. Math. Soc. 369 (2017), 575-603 Request permission

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:

  • [–] $\mathscr {WAP}_0(G)/C_0(G)$ 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.

  • [–] $\mathscr {WAP}(G)/\mathscr {B}(G)$, 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 $\kappa =\kappa (G)$.

  • [–] $\mathscr {WAP}_0(G)/\mathscr {B}_0(G)$, when $G$ and $\kappa$ are as in the foregoing item.

  • [–] $\mathscr {CB}(G)/\mathscr {LUC}(G)$, 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
    • MR Author ID: 292620
    • Email: mfilali@cc.oulu.fi
    • Jorge Galindo
    • Affiliation: Instituto Universitario de Matemáticas y Aplicaciones (IMAC), Universidad Jaume I, E-12071, Castellón, Spain
    • MR Author ID: 615222
    • Email: jgalindo@mat.uji.es
    • 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$\cdot$1B2014-35.
    • © Copyright 2016 American Mathematical Society
    • Journal: Trans. Amer. Math. Soc. 369 (2017), 575-603
    • MSC (2010): Primary 22D15; Secondary 43A46, 43A15, 43A60
    • DOI: https://doi.org/10.1090/tran6662
    • MathSciNet review: 3557786