On the Extension of Operators with Range in a C(k) Space

Theorem. For X= C(K) the following four statements are equivalent. (i) For every two Banach spaces ZD Y with dim(Z/ Y) = 1 and every operator T from Y into X with a separable range there is an extension T of T from Z into X with \\t\\=\\T\\. •J J 11 11 11 M (ii) For every two Banach spaces ZZ)Y with dim Y =2, dim Z = 3 and every operator T from Y into X there is an extension f of T from Z into X with ||f|| = ||r||. (iii) There is a X<2 such that for every two Banach spaces ZZ)Y with dim(Z/ Y) = 1 and every operator T from Y into X with a separable range there is an extension T of T from Z into X with || T|| ^\|| T\\. (iv) K is an F-space in the terminology of Gillman and Jerison [4, p. 208]. That is, for every fEC(K) there is a gEC(K) such thatf(k)>0 implies g(k) ^ 1 and f(k) <0 implies g(k) á — 1.