A renorming characterization of Banach spaces containing ℓ1(κ)
Avilés, Antonio (Universidad de Murcia. Departamento de Matemáticas)
Martínez-Cervantes, Gonzalo (Universidad de Alicante. Departamento de Matemáticas)
Rueda Zoca, Abraham (Universidad de Granada. Departamento de Análisis Matemático)

Data:	2023
Resum:	A result of G. Godefroy asserts that a Banach space X contains an isomorphic copy of '1 if and only if there is an equivalent norm ||| · ||| such that, for every finite-dimensional subspace Y of X and every ε > 0, there exists x ∈ SX so that |||y+rx||| ≥ (1−ε)(|||y|||+|r|) for every y ∈ Y and every r ∈ R. In this paper we generalise this result to larger cardinals, showing that if κ is an uncountable cardinal, then a Banach space X contains a copy of '1(κ) if and only if there is an equivalent norm ||| · ||| on X such that for every subspace Y of X with dens(Y ) < κ there exists a norm-one vector x so that |||y + rx||| = |||y||| + |r| whenever y ∈ Y and r ∈ R. This result answers a question posed by S. Ciaci, J. Langemets, and A. Lissitsin, where the authors wonder whether the above statement holds for infinite successor cardinals. We also show that, in the countable case, the result of Godefroy cannot be improved to take ε = 0.
Ajuts:	Agencia Estatal de Investigación MTM2017-86182-P Agencia Estatal de Investigación PGC2018-093794-B-I00
Nota:	Altres ajuts: Research partially supported by Fundación Séneca [20797/PI/18] and by project MTM2017-86182-P (Government of Spain, AEI/FEDER, EU). The research of G. Martínez-Cervantes was co-financed by the European Social Fund and the European Youth Initiative under Fundación Séneca [21319/PDGI/19]. The research of A. Rueda Zoca was also supported by Juan de la Cierva-Formaci'on fellowship FJC2019-039973, by MICINN (Spain) Grant PGC2018-093794-B-I00 (MCIU, AEI, FEDER, EU), by Junta de Andalucía Grant A-FQM-484-UGR18, and by Junta de Andalucía Grant FQM-0185.
Drets:	Tots els drets reservats.
Llengua:	Anglès
Document:	Article ; recerca ; Versió publicada
Matèria:	Renorming ; ℓ1(κ) ; Octahedral norm ; Ball-covering
Publicat a:	Publicacions matemàtiques, Vol. 67 Núm. 2 (2023) , p. 601-609 (Articles) , ISSN 2014-4350

Adreça original: https://raco.cat/index.php/PublicacionsMatematiques/article/view/418412
DOI: 10.5565/PUBLMAT6722305

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