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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.

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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.

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  • $\begingroup$ 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
    Commented Nov 19, 2016 at 19:07
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    $\begingroup$ 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. $\endgroup$
    – Mikhail Ostrovskii
    Commented Nov 19, 2016 at 19:15

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