On decompositions of Banach spaces of continuous functions on Mrówka’s spaces

Koszmider, Piotr

doi:10.1090/s0002-9939-05-07799-3

Cited by 32 publications

(43 citation statements)

References 15 publications

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“…In a recent paper P. Koszmider [8], under CH, has constructed a nonseparable C(K) space satisfying the property that whenever C(K) ∼ = Y ⊕ Z then either Y ∼ = c 0 and Z ∼ = C(K) or vice versa. In the same paper he also asked whether a separable Banach space could occur sharing similar properties.…”

Section: Introductionmentioning

confidence: 99%

Banach spaces with a unique nontrivial decomposition

Argyros

Raikoftsalis

2008

Proc. Amer. Math. Soc.

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Abstract. Motivated by a problem of P. Koszmider we introduce the class of quasi-prime Banach spaces. This class lies between the classes of prime and primary Banach spaces. It is shown that for every 1 < p < ∞ there exists a strictly quasi-prime separable reflexive Banach space X p such that p is a complemented subspace of X p . A similar result also holds for the case of 1 and c 0 . More generally, for every separable decomposable prime Banach space Y not containing 1 there exists a strictly quasi-prime X Y containing Y as a complemented subspace. We also investigate the operators acting on these spaces as well as the complemented subspaces of their finite powers.

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Section: Introductionmentioning

confidence: 99%

Banach spaces with a unique nontrivial decomposition

Argyros

Raikoftsalis

2008

Proc. Amer. Math. Soc.

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“…2 An example of a scattered K (of height three, but with C (K ) which is not Lindelöf) where all nontrivial decompositions of C (K ) have one factor isomorphic to the C (K ) and the other to c 0 was obtained under CH in [12]. Also Argyros and Raikoftsalis have constructed in [4] a separable Banach space X (necessarily not of the form C (K )) where all nontrivial decompositions are of the form c 0 ⊕ X .…”

Section: Claimmentioning

confidence: 99%

“…As shown in 4.11, one factor is isomorphic to c 0 or C 0 (ω ω ) and the other is isomorphic to C (K 0 ). This is related to few decompositions as in [12] and [4].…”

Section: Introductionmentioning

confidence: 99%

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Complementation and decompositions in some weakly Lindelöf Banach spaces

Koszmider

Zieliński

2011

Journal of Mathematical Analysis and Applications

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Let Γ 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 (Γ ) or (2) for every c 0 (Γ ) ⊆ X has a complemented c 0 (E) for an uncountable E ⊆ Γ or (3) has a decomposition X = A ⊕ B where both A and B are nonseparable. The results concern a superclass of the class of nonmetrizable Eberlein compacts, namely K s such that C (K ) is Lindelöf in the weak topology and we restrict our attention to K s 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 (Γ ) ⊆ C (K ) there is a complemented c 0 (E) for an uncountable E ⊆ Γ , which yields the positive answer to the remaining questions. If we assume ♣, then we construct a nonseparable weakly Lindelöf C (K ) for K of height ω + 1 where every operator is of the form cI + S for c ∈ 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 Lindelöf properties of these two topologies are equivalent, many results concern also the space C p (K ).

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“…In fact, a modification of a reasoning from [19,Lemma 4.3] shows that there is a collection M of almost disjoint families on ω, with |M| = 2 2 ℵ 0 , such that no two distinct spaces (C(K A )), weak), A ∈ M, are homeomorphic, cf. [17,Question 5].…”

Section: Topological Types Of Function Spaces Associated With Almost mentioning

confidence: 99%

On Banach spaces whose norm-open sets are Fσ-sets in the weak topology

Marciszewski

Pol

2009

Journal of Mathematical Analysis and Applications

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We investigate non-separable Banach spaces whose norm-open sets are countable unions of sets closed in the weak topology and a narrower class of Banach spaces with a network for the norm topology which is σ -discrete in the weak topology. In particular, we answer a question of Arhangel'skii exhibiting various examples of non-separable function spaces C(K) with a σ -discrete network for the pointwise topology and (consistently) we answer some questions of Edgar and Oncina concerning Borel structures and Kadec renormings in Banach spaces.

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On decompositions of Banach spaces of continuous functions on Mrówka’s spaces

Cited by 32 publications

References 15 publications

Banach spaces with a unique nontrivial decomposition

Banach spaces with a unique nontrivial decomposition

Complementation and decompositions in some weakly Lindelöf Banach spaces

On Banach spaces whose norm-open sets are Fσ-sets in the weak topology

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