Linear extension operators of bounded norms

Dugundji spaces were introduced by Pełczyński as compact Hausdorff spaces X such that every embedding of X into a Tychonoff cube [0,1]A admits a linear extension operator u:C(X)→C([0,1]A) such that ‖u‖=1 and u(1X)=1[0,1]A, where 1X is the constant function on X taking value 1. In this paper we show that a compact space X is Dugundji provided that there exists a linear extension operator u:C(X)→C([0,1]A) satisfying one of the following conditions: