A note on nonlinear isometries between vector-valued function spaces

Abstract

Let X, Y be compact Hausdorff spaces and E, F be Banach spaces over R or C. In this paper, we investigate the general form of surjective (not necessarily linear) isometries T : A −→ B between subspaces A and B of C(X, E) and C(Y, F ), respectively. In the case that F is strictly convex, it is shown that there exist a subset Y 0 of Y , a continuous function Φ : Y 0 −→ X onto the set of strong boundary points of A and a family {V y } y∈Y0 of real-linearIn particular, we get some generalizations of the vector-valued Banach-Stone theorem and a generalization of Cambern's result. We also give a similar result in the case that F is not strictly convex, but its unit sphere contains a maximal convex subset which is singleton.2010 Mathematics Subject Classification. Primary 46J10, 46J20; Secondary 47B33. Key words and phrases. isometries, vector-valued continuous functions, strong boundary points, Choquet boundary . 1 strong operator topology, such that T f (t) = V t (f (Φ(t)) for all f ∈ C(X, E) and t ∈ Y .This result has been generalized by Cambern [4] for into linear isometries and by Font [8] for certain vector-valued subspaces of continuous functions. In [1] Al-Halees and Fleming relaxed the strict convexity condition on E, and by considering T -sets in companion with another condition on E, called condition (P), they obtained more general results. We also refer the reader to the nice books [6,7] including many earlier results.More recently, surjective isometries between certain subspaces of vector-valued continuous functions have been studied in [3] and [11]. We should note that the method used in these papers is based on extreme point technique. By [3] (see also [2]), for a compact Hausdorff space X and a reflexive real Banach space F whose dual is strictly convex, if A is a subspace of C(X, F ) which separates X in the sense of [3, Definition 3.1], and T : A −→ A is a surjective isometry preserving constant functions, then there exist a surjective isometry V : F −→ F and a homeomorphism τ : X −→ X such that

Isometries between completely regular vector-valued function spaces

“…In particular, if E is a normed space whose unit sphere S(E) contains an element e with St(e) = {e}, then E has property (D). For an example of a non-strictly convex space E such that St(e) = {e} for some e ∈ S(E), see [12].…”

“…We should note that in a strictly convex normed space E, any maximal convex subset of the unit sphere of E is a singleton. However, in [12] there are some results for surjective supremum norm isometries between vector-valued spaces of continuous functions with values in a Banach space E whose unit sphere contains a maximal convex subset which is a singleton. Characterizatin of surjective isometries on spaces of vector-valued continuously differentiable functions with values in a finite-dimensional real Hilbert space can be found in [3].…”

Isometries on certain non-complete vector-valued function spaces

“…Cases (ii) and (iii) give some other generalizations in this topic. For example of a non-strictly convex normed space F satisfying the hypothesis of Case (i) see [7]. Indeed, there are infinitely many norms • on R 2 such that (R 2 , • ) is non-strictly convex and satisfies the hypothesis of Case(i).…”