On a variant of Tingley's problem for some function spaces

2020/06/16 by Leung, Chi-Wai, Ng, Chi-Keung, Wong, Ngai-Ching · 1 citation
#46B04 #46E15 #46E30 #46G12 #47B33 #47B49 #47B65 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2006.08944

Abstract

Let (Ω, \mathfrakA, μ) and (Γ, \mathfrakB, ν) be two arbitrary measure spaces, and p∈ [1,∞]. Set L^p(μ)₊^sp:= \f∈ L^p(μ): ‖f‖_p =1; f≥ 0 μ-a.e. \ i.e., the positive part of the unit sphere of L^p(μ). We show that every metric preserving bijection Φ: L^p(μ)₊^sp → L^p(ν)₊^sp can be extended (necessarily uniquely) to an isometric order isomorphism from L^p(μ) onto L^p(ν). A Lamperti form, i.e., a weighted composition like form, of Φ is provided, when (Γ, \mathfrakB, ν) is localizable (in particular, when it is σ-finite). On the other hand, we show that for compact Hausdorff spaces X and Y, if Φ is a metric preserving bijection from the positive part of the unit sphere of C(X) to that of C(Y), then there is a homeomorphism τ:Y→ X satisfying Φ(f)(y) = f(τ(y)) (f∈ C(X)₊^sp; y∈ Y).

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