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On dyadic spaces and almost Milyutin spaces. (English) Zbl 0833.47025

In order to determine whether a compact space \(K\) is an almost Milyutin space, it is necessary to study if an arbitrary continuous map from a Cantor cube \(2^m\) onto \(K\) can admit an averaging operator. In [Trans. Am. Math. Soc. 175, 195-208 (1973; Zbl 0251.46056)], S. Z. Ditor obtains a sufficient condition in order that the norm of an averaging operator for a continuous onto map \(\varphi: S\to T\), where \(S\) and \(T\) are compact spaces, has a lower bound. This condition is given in terms of the topological structure of \(T\) and the decomposition induced by \(\varphi\) on \(S\). Here we give another condition which relies only upon the topological structure of the space \(T\) and which allows us to apply Ditor’s theorem to an arbitrary continuous map from \(2^m\) onto \(K\). As an application of our results, two problems posed by A. Pełczyński in [Dissertationes Math. Warszawa 58 (1968; Zbl 0165.146)]are solved.

MSC:

47B38 Linear operators on function spaces (general)
46E15 Banach spaces of continuous, differentiable or analytic functions
54C30 Real-valued functions in general topology
54B10 Product spaces in general topology
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