Linear Surjective Isometries Between Vector-Valued Function Spaces

Kawamura, Kazuhiro

doi:10.1017/s1446788715000518

Cited by 8 publications

(5 citation statements)

References 31 publications

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“…This result has been generalized in several directions in [11]. We refer one of them which is related to our results.…”

Section: Introductionsupporting

confidence: 67%

“…For the application of the results we give next corollaries. As we noted before, the following (S3) condition has been considered in [11] in some results.…”

Section: Resultsmentioning

confidence: 99%

“…We refer one of them which is related to our results. First we state conditions (S3) and (M) introduced in [11] for a subspace A of C(X, E) where X is a compact Hausdorff space and E is a Banach space over C or R: (S3) For each x in the Choquet boundary Ch(A) of A, for each neighborhood U of x and for each u ∈ E there exists a function f ∈ A such that f ∞ = u , f (x) = u and f = 0 on X\U.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…[11, Theorem 3.4] Let X and Y be compact Hausdorff spaces and let E be a strictly convex reflexive real or complex Banach space. Assume that A and B are subspaces of C(X, E), respectively, containing constant functions and both satisfy conditions (S3) and (M).…”

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A note on nonlinear isometries between vector-valued function spaces

Jamshidi

Sady

2018

Linear and Multilinear Algebra

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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

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“…This result has been generalized in several directions in [11]. We refer one of them which is related to our results.…”

Section: Introductionsupporting

confidence: 67%

“…For the application of the results we give next corollaries. As we noted before, the following (S3) condition has been considered in [11] in some results.…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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