Filali, M., Galindo, J. (2017) Interpolation sets and the size of quotients of function spaces on a locally compact group. Transactions of the American Mathematical Society, 369 (1), 575-603. https://doi.org/10.1090/tran6662

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

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Author: Filali, Mahmoud1; Galindo, Jorge2
Organizations: 1Department of Mathematical Sciences, University of Oulu, Oulu, Finland
2Instituto Universitario de Matemáticas y Aplicaciones (IMAC), Universidad Jaume I, E-12071, Castellón, Spain
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.4 MB)
Language: English
Published: American Mathematical Society, 2017
Publish Date: 2019-04-08
Description:

# 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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Series: Transactions of the American Mathematical Society
ISSN: 0002-9947
ISSN-E: 1088-6850
ISSN-L: 0002-9947
Volume: 369
Pages: 575 - 603
DOI: 10.1090/tran6662
Funding: 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.