On universal spaces of the type 𝐶(𝑋)

Abstract

each nonuniversal C(X) with dim C(X) > 1 there is a two dimensional space which is not embeddable in C(X). A characterization of those Banach spaces that can be linearly isometrically embedded in C(X) for every infinite compact Hausdorff space X is also given. For a given Banach space B, BT denotes the closed unit sphere of the dual B' of B in the weak* topology. The set of extreme points of Br is denoted by EBT and its weak* closure is denoted by cl*EBT. If X is a compact Hausdorff space, e denotes the evaluation map of X into C(X)'. It is well known that e is a homeomorphism into C(X)' with the weak* topology and that EC(X)'=e(X)kJ-e(X) [2, p. 86]. If i: B-*C(X) is a linear isometry into, then the adjoint i' of i maps C(X) onto BT, and, in fact, i'(EC(X)r) DEBT [3, p. 441]. The following proposition shows that not all infinite compact spaces X have the property that C(X) is universal for the set of all separable Banach spaces.