An operator is said to be nice if its conjugate maps extreme points of the dual unit ball to extreme points. The classical Banach-Stone Theorem says that an isometry from a space of continuous functions on a compact Hausdorff space onto another such space is a weighted composition operator. One common proof of this result uses the fact that an isometry is a nice operator. We use extreme point methods and the notion of centralizer to characterize nice operators as operator weighted compositions on subspaces of spaces of continuous functions with values in a Banach space. Previous characterizations of isometries from a subspace M of Q,( Q, X) into Q>(K, Y) require Y to be strictly convex, but we are able to obtain some results without that assumption. Important use is made of a vector-valued version of the Choquet Boundary. We also characterize nice operators from one function module to another.2000 Mathematics subject classification: primary 46B04, 46E40.

A (not necessarily linear) mapping Φ from a Banach space X to a Banach space Y is said to be a 2-local isometry if for any pair x, y of elements of X, there is a surjective linear isometry T : X → Y such that T x = Φx and T y = Φ y. We show that under certain conditions on locally compact Hausdorff spaces Q , K and a Banach space E, every 2-local isometry on C 0 (Q , E) to C 0 (K , E) is linear and surjective. We also show that every 2-local isometry on p is linear and surjective for 1 p < ∞, p = 2, but this fails for the Hilbert space 2 .A (not necessarily linear) mapping Φ from a Banach space X to a Banach space Y is said to be a 2-local isometry if for any pair x, y of elements of X , there is a surjective linear isometry T : X → Y such that Φx = T x and Φ y = T y. The general question is whether Φ must itself be a surjective linear isometry. This type of problem is basic in that it asks whether a local assumption is enough to guarantee a more global conclusion. Early investigations along these lines involved derivations and automorphisms of operator algebras and were carried out by Kadison [8], Larson [9], and Larson and Sourour [10]. A set S of operators is called algebraically reflexive if S must contain every T which is local in this sense: given x in the domain, there is an S ∈ S such that T x = Sx. If the group G(X) of surjective linear isometries on X is algebraically reflexive, we will say that X is iso-reflexive. This language could also be applied to a pair (X, Y ) of Banach spaces if the isometries go from X to Y .Results concerning iso-reflexivity of certain operator algebras and function algebras have been obtained, about which [2,6,13], with their references, serve as a good introduction. In particular, Molnár and Zalar [12] showed that if Q is compact, Hausdorff, and first countable, then C (Q ) is iso-reflexive. Jarosz and Rao [6] extended this to the vector-valued case, proving that if Q is a first countable compact Hausdorff space and E is a uniformly convex and iso-reflexive Banach space, thenThe notion of 2-local is due to Šemrl [14] who was interested in dropping the linearity assumption for local automorphisms and derivations on L(H), the bounded linear operators on H , where H is a Hilbert space. To compensate for the loss of linearity, it was useful to require the local condition at two points. Molnár [11] showed that every 2-local isometry on L(H) is linear and so a surjective linear isometry. Gyory [5] showed that if Q is a first countable, σ -compact, (separable) locally compact Hausdorff space, then every 2-local isometry on C 0 (Q ) is a surjective linear isometry. That paper is the inspiration for the current note, in which we wish to consider the extension of Gyory's theorem to C 0 (Q , E) for an appropriate Banach space E. By C 0 (Q , E) we mean, of course, the continuous functions on Q to E which vanish at infinity and given the sup norm. In case E is the scalar field, we just write C 0 (Q ).Let us agree to say that a Banach space X is 2-iso-reflexive if every 2-local isometry...

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Abstract. An old question asks whether extreme contractions on C(K) are necessarily nice; that is, whether the conjugate of such an operator maps extreme points of the dual ball to extreme points. Partial results have been obtained. Determining which operators are extreme seems to be a difficult task, even in the scalar case. Here we consider the case of extreme contractions on C (K, E), where E itself is a Banach space. We show that every extreme contraction T on C(K, E) to itself which maps extreme points to elements of norm one is nice, where K is compact and E is the sequence space c 0 .By an extreme contraction, we mean an element T of the set L(X, Y ) of bounded linear operators from a Banach space X to a Banach space Y , which is an extreme point of the unit ball of L(X, Y ). In 1965, Blumenthal, Lindenstrauss, and Phelps [2] showed that the conjugate T * of an extreme contraction T on the space C(Q) of real-valued continuous functions on a compact metric space Q to C(K), where K is compact and Hausdorff, must map the extreme points of C(K) * to extreme points. Such operators have been called nice operators [6]. In fact, what Blumenthal, Lindenstrauss, and Phelps really showed was that the operator T must have the form T f(t) = h(t)f (ϕ(t)),where h ∈ C(K) with modulus one, and ϕ is continuous on K to Q. This form is typical of nice operators on continuous function spaces, even in the vector-valued case [1].It is always the case, and easy to show, that every nice operator is extreme. The general question to be asked, then, is the converse: if T ∈ L(X, Y ) is extreme, must it be nice? The answer, in general, is no, since it has been shown that there exists a four-dimensional Banach space X and an extreme contraction T from X to C [0,1] that is not nice [2]. However, there is a positive answer for certain C(K) spaces, as we have already mentioned above, and other versions of the BLP result have been given by several authors. Utilising results of Sharir [7], Gendler [3] extended the result of BLP [2] to the case of complex-valued functions as follows.Theorem 1 (Gendler). Let Q, K be compact Hausdorff spaces and let T be an extreme contraction from the space C(Q) of complex-valued functions on Q to the