Homogeneous families on trees and subsymmetric basic sequences

Autor(es)

Brech, C. ^[1] ; Lopez-Abad, J. ^{[2, 3]} ; Todorcevic, S. ^{[4, 5]}

Afiliação do(s) autor(es)

^[1] Univ Sao Paulo, Inst Matemat & Estat, Dept Matemat, Caixa Postal 66281, BR-05314970 Sao Paulo - Brazil

^[2] Univ Paris 07, Inst Mathemat Jussieu PRG, Case 7012, F-75205 Paris 13 - France

^[3] UNED, Dept Matemat Fundamentales, Paseo Senda Rey 9, E-28040 Madrid - Spain

^[4] Univ Toronto, Dept Math, Toronto, ON M5S 2E4 - Canada

^[5] Inst Math Jussieu, UMR 7586, 2 Pl Jussieu Case 247, F-75222 Paris 05 - France

• Tipo de documento: Artigo Científico
• Fonte: ADVANCES IN MATHEMATICS; v. 334, p. 322-388, AUG 20 2018.
• Citações Web of Science: 0

Resumo

We study density requirements on a given Banach space that guarantee the existence of subsymmetric basic sequences by extending Tsirelson's well-known space to larger index sets. We prove that for every cardinal a smaller than the first Mahlo cardinal there is a reflexive Banach space of density a without subsymmetric basic sequences. As for Tsirelson's space, our construction is based on the existence of a rich collection of homogeneous families on large index sets for which one can estimate the complexity on any given infinite set. This is used to describe detailedly the asymptotic structure of the spaces. The collections of families are of independent interest and their existence is proved inductively. The fundamental stepping up argument is the analysis of such collections of families on trees. (C) 2018 Elsevier Inc. All rights reserved. (AU)

• Processo FAPESP: 15/26654-2 - Geometria de espaços de Banach
• Beneficiário: Christina Brech
• Modalidade de apoio: Bolsas no Exterior - Pesquisa
• Processo FAPESP: 12/24463-7 - Métodos combinatórios em Espaços de Banach
• Beneficiário: Christina Brech
• Modalidade de apoio: Auxílio à Pesquisa - Regular
• Processo FAPESP: 13/24827-1 - Métodos de teoria de Ramsey em espaços de Banach
• Beneficiário: Valentin Raphael Henri Ferenczi
• Modalidade de apoio: Auxílio à Pesquisa - Pesquisador Visitante - Internacional