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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On a lemma of Milutin concerning averaging operators in continuous function spaces

Author: Seymour Z. Ditor
Journal: Trans. Amer. Math. Soc. 149 (1970), 443-452
MSC: Primary 47B37
MathSciNet review: 0435921
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Abstract: We show that any infinite compact Hausdorff space S is the continuous image of a totally disconnected compact Hausdorff space $ S'$, having the same topological weight as S, by a map $ \varphi $ which admits a regular linear operator of averaging, i.e., a projection of norm one of $ C(S')$ onto $ {\varphi ^ \circ }C(S)$, where $ {\varphi ^\circ }:C(S) \to C(S')$ is the isometric embedding which takes $ f \in C(S)$ into $ f \circ \varphi $. A corollary of this theorem is that if S is an absolute extensor for totally disconnected spaces, the space $ S'$ can be taken to be the Cantor space $ {\{ 0,1\} ^\mathfrak{m}}$, where $ \mathfrak{m}$ is the topological weight of S. This generalizes a result due to Milutin and Pełczyński. In addition, we show that for compact metric spaces S and T and any continuous surjection $ \varphi :S \to T$, the operator $ u:C(S) \to C(T)$ is a regular averaging operator for $ \varphi $ if and only if u has a representation $ uf(t) = \smallint _0^1f(\theta (t,x))$ for a suitable function $ \theta :T \times [0,1] \to S$.

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Keywords: Averaging operator, extension operator, linear exave, projection operator, regular operator, continuous function space, Milutin space, totally disconnected space, inverse and direct limits, integral representation of regular averaging and extension operators
Article copyright: © Copyright 1970 American Mathematical Society