Uniform homeomorphisms of unit spheres and Property H of Lebesgue–Bochner function spaces
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- Cheng, Q.J. & Dong, Y.B. Acta. Math. Sin.-English Ser. (2017) 33: 681. doi:10.1007/s10114-016-6245-1
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Assume that the unit spheres of Banach spaces X and Y are uniformly homeomorphic. Then we prove that all unit spheres of the Lebesgue–Bochner function spaces Lp(μ,X) and Lq(μ, Y) are mutually uniformly homeomorphic where 1 ≤ p, q < ∞. As its application, we show that if a Banach space X has Property H introduced by Kasparov and Yu, then the space Lp(μ,X), 1 ≤ p < ∞, also has Property H.