Smooth Norms and Approximation in Banach Spaces of the Type C(k)

We prove two results about smoothness in Banach spaces of the type C(K). Both build upon earlier papers of the first named author [3,4]. We first establish a special case of a conjecture that remains open for general Banach spaces and concerns smooth approximation. We recall that a bump function on a Banach space X is a function β : X → R which is not identically zero, but which vanishes outside some bounded set. The existence of bump function of class C implies that the Banach space X is an Asplund space, which, in the case where X = C(K), is the same as saying that K is scattered. It is a major unsolved problem to determine whether every Asplund space has a C bump function. Another open problem is whether the existence of just one bump function of some class C on a Banach space X implies that all continuous functions on X may be uniformly approximated by functions of class C. It is to this question that we give a positive answer (Theorem 2) in the special case of X = C(K). Our second result represents some mild progress with a conjecture made by the second author in [7]. The analysis in that paper of compact spaces constructed using trees suggested that for a compact space K the existence of an equivalent norm on C(K) which is of class C (except at 0 of course) might imply the existence of such a norm which is of class C∞. Certainly, this is what happens with norms constructed using linear Talagrand operators as in [5,6,7]. The other important (and older) method of obtaining C norms is to construct a norm with locally uniformly rotund dual norm. What we show in Theorem 2 is that, whenever C(K) admits an equivalent norm with LUR dual norm, then there is also an infinitely differentiable equivalent norm on C(K). For background on smoothness and renormings in Banach spaces, including an account of Asplund spaces, we refer the reader to [1]. In particular, an account is given there of the connection between smooth approximability of continuous functions and the existence of smooth partitions of unity. Following what seems to be standard practice in the literature, we have chosen to state the formal version of our first theorem (Theorem 1) in terms of partitions of unity, rather than approximation.