Abstract
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.
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