Linear extension operators of bounded norms (Q1645178)

Recall that a compact Hausdorff space \(X\) is said to be a Dugundji space if, for any embedding \(e\) of \(X\) into another compact Hausdorff space \(Y\), there exists a regular extension operator \(u: C(X) \to C(Y)\), that is, \(u(f) \circ e= f\), for every \(f \in C(X)\), \(\|u\|=1\) and \(u(1_{X})=1_{Y}\), where \(1_{X}\) and \(1_{Y}\) are the constant functions on \(X\) and \(Y\), respectively, taking the value 1, see [\textit{A. Pełczyński}, Diss. Math. 58, 92 p. (1968; Zbl 0165.14603)]. The authors consider the following natural question. Is it true that a space \(X\) is a Dugundji space if there exist an embedding of \(X\) into a Tychonoff space \([0,1]^A\) and a linear extension operator \(u:C(X) \to C([0,1]^A)\) satisfying \(\|u\|<2\)? First, they give a positive answer to this question, when for any \(y \in Y\) with \(u(1_X)(y)=1\) and \(f, g \in C(X)\), we have \(|u(fg)(y)| \leq \|g\| \;|u(|f|)(y)|\). Then, they present some consequences of this result. Other interesting problems related to Dugundji spaces are left open.