10
$\begingroup$

While it is known that compact Hausdorff spaces admit unique uniform structures, it is further shown by Johson and Lindenstrauss's result that Banach spaces are characterized by their uniform structures [Johson&Lindenstrauss&Schechtman]. [Ribe] pointed out that two non-isomorphic spaces may have homeomorphic uniform structures (See [Benyamini&Lindenstrauss, Chap 10, Sec 10.4]).

The classical nonunique example in [Benyamini&Lindenstrauss, Chap 10, Prop 10.32-10.34] shown that p-convexified Tsirelson spaces $\mathcal{T}^{(p)}$ for $1<p<\infty$ admit exactly two different uniform structures.

Is there an example that a Banach space $X$ admits more than two uniform structures?

[Johson&Lindenstrauss&Schechtman]Johson, W. B., Joram Lindenstrauss, and Gideon Schechtman. "Banach spaces determined by their uniform structures." Geometric & Functional Analysis GAFA 6.3 (1996): 430-470, doi: 10.1007/BF02249259, arXiv:math/9701203.

[Benyamini&Lindenstrauss]Benyamini, Yoav, and Joram Lindenstrauss. Geometric nonlinear functional analysis. Vol. 48. American Mathematical Soc., 1998.

[Ribe]Ribe, Martin. "On uniformly homeomorphic normed spaces." Arkiv för matematik 14.1-2 (1976): 237-244, doi: 10.1007/BF02385837.

Improve this question
$\endgroup$
6
  • 1
    $\begingroup$ What do you mean by a space -- a topological space? If yes, the answer there is extremely well-known: under the standard metric, the open interval $(0, 1)$ is not complete, but under another (transporting the standard metric on $\mathbb{R}$ across a homeomorphism $\mathbb{R} \to (0, 1)$) it is, so these uniform structures are inequivalent. $\endgroup$
    – Todd Trimble ♦
    May 8, 2017 at 0:19
  • 1
    $\begingroup$ @ToddTrimble What I have in mind is a complete topological space, i.e. its Cauchy filters converges. You are right about the $(0,1)$ example, I have edited the OP to avoid this trivial example. Thanks for the input! $\endgroup$
    – Henry.L
    May 8, 2017 at 0:23
  • 4
    $\begingroup$ Unless I am gravely mistaken, Cauchy filters are defined for uniform spaces, but not for general topological spaces. en.wikipedia.org/wiki/Filter_(mathematics)#Cauchy_filters $\endgroup$
    – Todd Trimble ♦
    May 8, 2017 at 0:33
  • $\begingroup$ @ToddTrimble You are right about this point too, I wanted to ask for Banach space at the beginning and generalize the question a bit but obivously not successful. And this post is caused by my ignorance of one of the results contained in Johnson's paper as he pointed out below. It is fine to close it. Have corrected it. $\endgroup$
    – Henry.L
    May 8, 2017 at 0:37
  • 2
    $\begingroup$ Mazur's homeomorphisms are uniform on bounded sets in both directions, but not on the entire space. $\endgroup$
    – Bill Johnson
    May 8, 2017 at 14:03

1 Answer 1

Reset to default
10
$\begingroup$

In Theorem 5.8 of my paper with Lindenstrauss and Schechtman that you reference we show that for every $n=0,1,2,\dots$ there exists a Banach space that is uniformly homeomorphic to exactly $2^n$ mutually non isomorphic spaces. AFAIK, it is still open whether $2^n$ can be replace by any number that is not a power of $2$. See (5) at the end of the paper.

Improve this answer
$\endgroup$
1
  • $\begingroup$ It is a lesson to me that I should never skip each result contained in an important paper, thank you so much. I was expecting an arbitrary number $\alpha$ but Thm 5.8 is good enough for me. Thanks again! $\endgroup$
    – Henry.L
    May 8, 2017 at 0:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.