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 L p (μ,X) and L q (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 L p (μ,X), 1 ≤ p < ∞, also has Property H.
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Acknowledgements
The first author is grateful to Professor Guoliang Yu and Professor Qin Wang for bringing attention to some problems of Property H. Part of this work was done during the first author’s visit to the Department of Mathematics at Texas A&M University. The authors thank Professor Lixin Cheng for his valuable suggestion and helpful discussion and thank people in the Functional Analysis Seminar of Xiamen University. We also thank the referees for their comments.
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The first author is supported by National Natural Science Foundation of China (Grant No. 11471271); the second author is supported by the Foundation of Hubei Provincial Department of Education (Grant No. Q20161602)
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Cheng, Q.J., Dong, Y.B. Uniform homeomorphisms of unit spheres and Property H of Lebesgue–Bochner function spaces. Acta. Math. Sin.-English Ser. 33, 681–690 (2017). https://doi.org/10.1007/s10114-016-6245-1
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DOI: https://doi.org/10.1007/s10114-016-6245-1