5
$\begingroup$

I wonder if the following assertion in true:

Conjecture. Let $X,Y,Z$ be infinite-dimensional Banach spaces such that both $Y$ and $Z$ are crudely finitely representable (c.f.r. for short) in $X$. Then $Y\oplus Z$ is c.f.r. in $X$.

Remark some equivalent formulations of the above conjecture.

(A) For every infinite-dimensional Banach space $X$, $X\oplus X$ is c.f.r. in $X$.

(B) For every infinite-dimensional Banach space $X$, $X\oplus X$ is isomorphic to a subspace of an ultrapower of $X$.

I guess this question could be connected with those of whether a Banach space isomorphic to its square (solved in the negative by Figiel and later improved by Gowers). But perhaps it is much simpler.

Thanks in advance.

$\endgroup$

1 Answer 1

7
$\begingroup$

The conjecture which you stated is false. A counterexample is contained in the proof of Figiel [Studia Math. 42 (1972), 295–306]. He actually proves that squares of finite-dimensional subspaces of the space he constructs are not uniformly embeddable into the space itself.

I am unaware of a simpler counterexample for infinite-dimensional spaces.

$\endgroup$
2
  • $\begingroup$ Thanks! Let me ask you the following weaker conjecture: For any Banach space $X$, $X\oplus \ell_2$ is c.f.r. in $X$. $\endgroup$
    – Anso
    Nov 19, 2016 at 19:07
  • 1
    $\begingroup$ This weaker conjecture is true because each finite-dimensional subspace of $X\oplus \ell_2$ is close to being a subspace of $F\oplus \ell_2^n$ for some finite-dimensional subspace $F$ of $X$ and some $n\in\mathbb{N}$. Then you have to use the Dvoretzky theorem and a kind of the Mazur's argument (See Lindenstrauss-Tzafriri, v. I, Lemma 1.a.6) to show that $F\oplus \ell_2^n$ admits a linear embedding into $X$ with distortion bounded by an absolute constant. $\endgroup$ Nov 19, 2016 at 19:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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