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A simultaneous selection theorem

100%
Arvanitakis A.
Fundamenta Mathematicae
|
2012
|
tom 219
|
nr 1
1-14
EN
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.
2
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Some remarks on Radon-Nikodym compact spaces

100%
Arvanitakis A.
Fundamenta Mathematicae
|
2002
|
tom 172
|
nr 1
41-60
EN
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 space which is almost totally disconnected is actually a Radon-Nikodým compact space embeddable in the space of probability measures on a scattered compact space.
3
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A characterization of regular averaging operators and its consequences

63%
Argyros S., Arvanitakis A.
Studia Mathematica
|
2002
|
tom 151
|
nr 3
207-226
EN
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 a new characterization of Eberlein compact sets.
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