2000 Mathematics Subject Classification: Primary 46E15, 54C55; Secondary 28B20.
For weakly compact subsets of Hilbert spaces K, we study the
existence of totally disconnected spaces L, such that C(K) is isomorphic
to C(L).
We prove that the space C(BH ) admits a Pełczyński decomposition and
we provide a starshaped weakly compact K, subset of BH with non-empty
interior in the norm topology, and such that C(K) ~= C(L) with L totally disconnected.
Research partially supported...
The class of quasi Radon-Nikodým compact spaces is introduced. We prove that this class is closed under countable products and continuous images. It includes the Radon-Nikodým compact spaces. Adapting Alster's proof we show that every quasi Radon-Nikodým and Corson compact space is Eberlein. This generalizes earlier results by J. Orihuela, W. Schachermayer, M. Valdivia and C. Stegall. Further the class of almost totally disconnected spaces is defined and it is shown that every quasi Radon-Nikodým...
We prove a theorem that generalizes in a way both Michael's Selection Theorem and Dugundji's Simultaneous Extension Theorem. We use it to prove that if K is an uncountable compact metric space and X a Banach space, then C(K,X) is isomorphic to C(𝓒,X) where 𝓒 denotes the Cantor set. For X = ℝ, this gives the well known Milyutin Theorem.
We present a characterization of continuous surjections, between compact metric spaces, admitting a regular averaging operator. Among its consequences, concrete continuous surjections from the Cantor set 𝓒 to [0,1] admitting regular averaging operators are exhibited. Moreover we show that the set of this type of continuous surjections from 𝓒 to [0,1] is dense in the supremum norm in the set of all continuous surjections. The non-metrizable case is also investigated. As a consequence, we obtain...
We provide a characterization of continuous images of Radon-Nikodým compacta lying in a product of real lines and model on it a method for constructing natural examples of such continuous images.
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