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A function space Cp(X) not linearly homeomorphic to Cp(X) × ℝ

Witold Marciszewski — 1997

Fundamenta Mathematicae

We construct two examples of infinite spaces X such that there is no continuous linear surjection from the space of continuous functions c p ( X ) onto c p ( X ) × ℝ . I n p a r t i c u l a r , cp(X) i s n o t l i n e a r l y h o m e o m o r p h i c t o cp(X) × . One of these examples is compact. This answers some questions of Arkhangel’skiĭ.

On Banach spaces C(K) isomorphic to c₀(Γ)

Witold Marciszewski — 2003

Studia Mathematica

We give a characterization of compact spaces K such that the Banach space C(K) is isomorphic to the space c₀(Γ) for some set Γ. As an application we show that there exists an Eberlein compact space K of weight ω ω and with the third derived set K ( 3 ) empty such that the space C(K) is not isomorphic to any c₀(Γ). For this compactum K, the spaces C(K) and c ( ω ω ) are examples of weakly compactly generated (WCG) Banach spaces which are Lipschitz isomorphic but not isomorphic.

Modifications of the double arrow space and related Banach spaces C(K)

Witold Marciszewski — 2008

Studia Mathematica

We consider the class of compact spaces K A which are modifications of the well known double arrow space. The space K A is obtained from a closed subset K of the unit interval [0,1] by “splitting” points from a subset A ⊂ K. The class of all such spaces coincides with the class of separable linearly ordered compact spaces. We prove some results on the topological classification of K A spaces and on the isomorphic classification of the Banach spaces C ( K A ) .

Some remarks on universality properties of / c

Mikołaj KrupskiWitold Marciszewski — 2012

Colloquium Mathematicae

We prove that if is not a Kunen cardinal, then there is a uniform Eberlein compact space K such that the Banach space C(K) does not embed isometrically into / c . We prove a similar result for isomorphic embeddings. Our arguments are minor modifications of the proofs of analogous results for Corson compacta obtained by S. Todorčević. We also construct a consistent example of a uniform Eberlein compactum whose space of continuous functions embeds isomorphically into / c , but fails to embed isometrically....

Extension operators on balls and on spaces of finite sets

Antonio AvilésWitold Marciszewski — 2015

Studia Mathematica

We study extension operators between spaces of continuous functions on the spaces σ ( 2 X ) of subsets of X of cardinality at most n. As an application, we show that if B H is the unit ball of a nonseparable Hilbert space H equipped with the weak topology, then, for any 0 < λ < μ, there is no extension operator T : C ( λ B H ) C ( μ B H ) .

A contribution to the topological classification of the spaces Ср(X)

Robert CautyTadeusz DobrowolskiWitold Marciszewski — 1993

Fundamenta Mathematicae

We prove that for each countably infinite, regular space X such that C p ( X ) is a Z σ -space, the topology of C p ( X ) is determined by the class F 0 ( C p ( X ) ) of spaces embeddable onto closed subsets of C p ( X ) . We show that C p ( X ) , whenever Borel, is of an exact multiplicative class; it is homeomorphic to the absorbing set Ω α for the multiplicative Borel class M α if F 0 ( C p ( X ) ) = M α . For each ordinal α ≥ 2, we provide an example X α such that C p ( X α ) is homeomorphic to Ω α .

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