COMPLEX EXTREMAL STRUCTURE IN SPACES OF CONTINUOUS-FUNCTIONS

This paper considers the space Y = C(T, X) of all continuous and bound ed functions from a topological space T to a complex normed space X wi th the sup norm. The extremal structure of the closed unit ball B(Y) i n Y has been intensively studied when X is strictly convex, that is, i n terms of its unitary functions (mappings from T into the unit sphere of X). We prove that if T is completely regular and X has finite dime nsion, then every function in B(Y) is expressible as a convex combinat ion of three unitary functions if and only if the condition dim T < di m X-R is satisfied (where dim T is the covering dimension of T and X-R denotes X considered as a real normed space). If X is infinite-dimens ional the above decomposition is always possible without restrictions about T. These results are interesting when X is complex strictly conv ex. As a consequence we state a surprising fact: The identity function on the unit ball of an infinite-dimensional complex normed space can be expressed as the average of three retractions of the unit ball onto the unit sphere. Really, such a representation is the best possible. (C) 1997 Academic Press.