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

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 $\theta :T \times [0,1] \to S$.