LAUSR

Abstract Dugundji spaces were introduced by Pelczynski 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 ( 1 X ) = 1 [ 0 , 1 ] A , where 1 X 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: (a) ‖ u ‖ 2 and | u ( f ⋅ g ) | ≤ ‖ g ‖ ⋅ | u ( | f | ) | for all f , g ∈ C ( X ) ; (b) ‖ u ‖ = 1 .