Author 
Candido, Leandro
Koszmider, Piotr 
Abstract  Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(Kn) or equivalently the nfold injective tensor product (circle times) over cap Cn(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 Cn(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 Yepsilon 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 Kn of weight omega(1) such that C(K) is Lindelof in the weak topology, C(Kn) contains a copy of c(0)(omega(1)), C(Kn(n)) does not contain a complemented copy of c(0)(omega(1)) while C(Kn(n+1)) does contain a complemented copy of c(0)(omega(1)). This shows that additional settheoretic 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 halfpcc 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 SpacesBanachSpaces Sequential Convergence 
Language  English 
Sponsor 
FAPESP [2013/207036]
grant PVE Ciencia sem Fronteiras  CNPq [406239/20134] 
Grant number 

Date  2016 
Published in  Studia Mathematica. Warszawa, v. 233, n. 3, p. 209226, 2016. 
ISSN  00393223 (Sherpa/Romeo, impact factor) 
Publisher  Associacao Paulista Medicina 
Extent  209226 
Origin 

Access rights  Closed access 
Type  Article 
Web of Science ID  WOS:000383524600002 
URI  http://repositorio.unifesp.br/handle/11600/49403 
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