Into Isometries ofC0(X,E)s

Jeang, Jyh-Shyang; Wong, Ngai–Ching

doi:10.1006/jmaa.1997.5267

Cited by 8 publications

(2 citation statements)

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“…where the unit ball B(E * ) is endowed with the weak * -topology [7]. It is not difficult to show that the Choquet Boundary, ch(M), of M as a subspace of C 0 (Q × B(E * )) is the set of all pairs (s, x * ), where s ∈ Q and x * is an extreme point of the unit ball of E * , denoted by ext(E * ).…”

Section: Remarkmentioning

confidence: 99%

On 2-local isometries on continuous vector-valued function spaces

Al-Halees

Fleming

2009

Journal of Mathematical Analysis and Applications

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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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Section: Remarkmentioning

confidence: 99%

On 2-local isometries on continuous vector-valued function spaces

Al-Halees

Fleming

2009

Journal of Mathematical Analysis and Applications

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“…Furthermore, ψ is a homeomorphism of Y onto X. As in the scalar-valued case, Jerison's results have been extended in many directions (see e.g., [5], [1], [15] or [6]). In particular, M. Cambern obtained in [8] the following formulation of Holsztyński's theorem for spaces of continuous vector-valued functions.…”

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confidence: 99%