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 , having the same topological weight as *S*, by a map which admits a regular linear operator of averaging, i.e., a projection of norm one of onto , where is the isometric embedding which takes into . A corollary of this theorem is that if *S* is an absolute extensor for totally disconnected spaces, the space can be taken to be the Cantor space , where 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 , the operator is a regular averaging operator for if and only if *u* has a representation for a suitable function .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1970-0435921-2

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