. FA ] 4 J ul 1 99 7 DIAMETER PRESERVING LINEAR BIJECTIONS OF C ( X )

The aim of this paper is to solve a linear preserver problem on the function algebra C(X). We show that in case X is a first countable compact Hausdorff space, every linear bijection φ : C(X) → C(X) having the property that diam(φ(f)(X)) = diam(f(X)) (f ∈ C(X)) is of the form φ(f) = τ · f ◦ φ+ t(f)1 (f ∈ C(X)) where τ ∈ C, |τ | = 1, φ : X → X is a homeomorphism and t : C(X) → C is a linear functional. Linear preserver problems concern the question of determing all linear maps on an algebra which leave a given set, function or relation defined on the underlying algebra invariant. The study of linear preserver problems on matrix algebras represents one of the most active research areas in matrix theory (see the survey paper [LiTs]). In the last decade a considerable attention has been paid to the infinite dimensional case as well and the investigations have resulted in several important results (e.g. [BrSe2]). Concerning function algebras, the main linear preserver problems studied so far are the characterizations of linear bijections preserving some given norm, respectively disjointness of the support (these latter maps are also called separating). The linear bijections of C(X) preserving the sup-norm are determined in the famous Banach-Stone theorem. To mention some recent papers, we refer to [Wan], [Wea], [FoHe] and [HBN]. One way of measuring a function f ∈ C(X) is to consider its sup-norm. One of the other several possibilities that have sense is to measure the function in question by some data which reflects how large its range is. For example, this can be done by considering the diameter of the range. The aim of this paper is to determine all linear bijections of C(X) which preserve the seminorm f 7→ diam(f(X)). For short, we call these maps diameter preserving. Obviously, every automorphism f 7→ f ◦φ (φ is a homeomorphism) of C(X) is diameter preserving. The diameter of sets in C is clearly invariant under rotation and translation. This gives us that every linear map of Date: July 3, 1997. 1991 Mathematics Subject Classification. Primary: 46J10, 47B38.