In this paper we first consider a real-linear isometry T from a certain subspace A of C (X ) (endowed with supremum norm) into C (Y ) where X and Y are compact Hausdorff spaces and give a result concerning the description of T whenever A is a uniform algebra on X . The result is improved for the case where T (A) is, in addition, a complex subspace of C (Y ). We also give a similar description for the case where A is a function space on X and the range of T is a real subspace of C (Y ) satisfying a ceratin separating property. Next similar results are obtained for real-linear isometries between spaces of Lipschitz functions on compact metric spaces endowed with a certain complete norm. MSC:46J10, 46J20, 47B48

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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

Let X, Y be locally compact Hausdorff spaces, A be a complex subspace of C 0 (X) and T : A → C 0 (Y ) be a real-linear isometry, whose range is not assumed to be a complex subspace of C 0 (Y ). In this paper, using the set Θ(A) and τ (A) consisting of all extremely strong boundary points and strong boundary points of A, respectively we introduce appropriate subsets Y 0 and Y 1 of Y and give a description of T on these sets. More precisely, we show that there exist continuous functions Φ :for all f ∈ A and y ∈ Y 0 . The result is improved in the case where either i) T (A) is a complex subspace of C 0 (Y ) and Θ(A) = ch(A), where ch(A) is the Choquet boundary of A or ii) T (A) satisfies a certain separating property. In the first case we show that there exists a clopen subset K of Y 0 such thatfor each f ∈ A and y ∈ Y 0 . In the second case we obtain similar results for τ (A) ∩ ch(A) and Y 1 instead of Θ(A) and Y 0 .

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