On complemented copies of c(0)(omega(1)) in c(k-n) spaces

On complemented copies of c(0)(omega(1)) in c(k-n) spaces

Author Candido, Leandro Autor UNIFESP Google Scholar
Koszmider, Piotr Google Scholar
Abstract Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(K-n) or equivalently the n-fold injective tensor product (circle times) over cap C-n(epsilon)(K) or the Banach space of vector valued continuous functions C(K,C(K,C(K...,C(K)...). We address the question of the existence of complemented copies of c(0)(omega(1)) in (circle times) over cap C-n(epsilon)(K) under the hypothesis that C(K) contains an isomorphic copy of c(0)(omega(1)). This is related to the results of E. Saab and P. Saab that X (circle times) over cap Y-epsilon contains a complemented copy of c(0), if one of the infinite dimensional Banach spaces X or Y contains a copy of c(0) and of E. M. Galego and J. Hagler that it follows from Martin's Maximum that if C(K) has density omega(1) and contains a copy of c(0)(omega(1)), then C(KXK) contains a complemented copy c(0)(omega(1)). Our main result is that under the assumption of (sic) for every n is an element of N there is a compact Hausdorff space K-n of weight omega(1) such that C(K) is Lindelof in the weak topology, C(K-n) contains a copy of c(0)(omega(1)), C(K-n(n)) does not contain a complemented copy of c(0)(omega(1)) while C(K-n(n+1)) does contain a complemented copy of c(0)(omega(1)). This shows that additional set-theoretic assumptions in Galego and Hagler's nonseparable version of Cembrano and Freniche's theorem are necessary as well as clarifies in the negative direction the matter unsettled in a paper of Dow, Junnila and Pelant whether half-pcc Banach spaces must be weakly pcc.
Keywords Banach Spaces Of Continuous Functions
Injective Tensor Product
Complemented Subspaces
Ostaszewski's (sic)
Martin's Maximum
Vector Valued Continuous Functions
Scattered Compact SpacesBanach-Spaces
Sequential Convergence
Language English
Sponsor FAPESP [2013/20703-6]
grant PVE Ciencia sem Fronteiras - CNPq [406239/2013-4]
Grant number FAPESP: [2013/20703-6
CNPq: 406239/2013-4
Date 2016
Published in Studia Mathematica. Warszawa, v. 233, n. 3, p. 209-226, 2016.
ISSN 0039-3223 (Sherpa/Romeo, impact factor)
Publisher Associacao Paulista Medicina
Extent 209-226
Origin http://dx.doi.org/10.4064/sm8181-4-2016
Access rights Closed access
Type Article
Web of Science ID WOS:000383524600002
URI http://repositorio.unifesp.br/handle/11600/49403

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