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Banach spaces determined by their uniform structures

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Abstract

Following results of Bourgain and Gorelik we show that the spaces ℓ p , 1<p<∞, as well as some related spaces have the following uniqueness property: IfX is a Banach space uniformly homeomorphic to one of these spaces then it is linearly isomorphic to the same space. We also prove that if aC(K) space is uniformly homeomorphic toc 0, then it is isomorphic toc 0. We show also that there are Banach spaces which are uniformly homeomorphic to exactly 2 isomorphically distinct spaces.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, 77843-3368, College Station, TX, USA

    W. B. Johson

  2. Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

    J. Lindenstrauss

  3. Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot, Israel

    G. Schechtman

Authors
  1. W. B. Johson
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  2. J. Lindenstrauss
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  3. G. Schechtman
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Additional information

Dedicated to the memory of E. Gorelik

The first author was an Erna and Jacob Michael Visiting Professor, The Weizmann Institute, 1994; and partially supported by NSF DMS 93-06376 and the U.S.-Israel Binational Science Foundation. The second and third authors were participants in the Workshop in Linear Analysis and Probability, Texas A&M University; and were partially supported by the U.S.-Israel Binational Science Foundation.

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Johson, W.B., Lindenstrauss, J. & Schechtman, G. Banach spaces determined by their uniform structures. Geometric and Functional Analysis 6, 430–470 (1996). https://doi.org/10.1007/BF02249259

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