Complementation and decompositions in some weakly Lindelof Banach spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:376

Article

Koszmider, Piotr^1,2  Zielinski, Przemyslaw^1,3 

[1] Tech Univ Lodz, Inst Math, PL-90924 Lodz, Poland

[2] Univ Granada, Fac Ciencias, Dept Anal Matemat, E-18071 Granada, Spain

[3] Polish Acad Sci, Inst Math, PL-00956 Warsaw, Poland

关键词: Banach spaces of continuous functions;    Weak Lindelof property;    Weak topology;    Linear operators;    Complemented subspaces;    Continuum hypothesis;    Scattered spaces;   

DOI  :  10.1016/j.jmaa.2010.11.014

来源: Elsevier

【 摘 要 】

Let Gamma denote an uncountable set. We consider the questions if a Banach space X of the form C(K) of a given class (1) has a complemented copy of c(0)(Gamma) or (2) for every c(0)(Gamma) subset of X has a complemented c(0)(E) for an uncountable E subset of Gamma or (3) has a decomposition X = A circle plus B where both A and B are nonseparable. The results concern a superclass of the class of nonmetrizable Eberlein compacts, namely Ks such that C(K) is Lindelof in the weak topology and we restrict our attention to Ks scattered of countable height. We show that the answers to all these questions for these C(K)s depend on additional combinatorial axioms which are independent of ZFC +/- CH. If we assume the P-ideal dichotomy, for every c(0)(Gamma) subset of C(K) there is a complemented c(0)(E) for an uncountable E subset of Gamma, which yields the positive answer to the remaining questions. If we assume (sic), then we construct a nonseparable weakly Lindelof C(K) for K of height omega + 1 where every operator is of the form cl + S for c is an element of R and S with separable range and conclude from this that there are no decompositions as above which yields the negative answer to all the above questions. Since, in the case of a scattered compact K, the weak topology on C(K) and the pointwise convergence topology coincide on bounded sets, and so the Lindelof properties of these two topologies are equivalent, many results concern also the space C-p(K). (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free