Let $X$ and $Y$ be Banach spaces and denote by $X\hat{\otimes}_\pi Y$ the projective tensor product.
Question: If $X\hat{\otimes}_\pi Y$ contains an isomorphic copy of $c_0$, must then $X$ or $Y$ contain an isomorphic copy of $c_0$ also?
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The answer is no. Bourgain and Pisier have given a counterexample (A construction of $\mathcal{L}_\infty$-spaces and related Banach spaces. Bol. Soc. Bras. Mat. 14, No. 2, 109-123 (1983). See Zbl 0586.46011 https://zbmath.org/?q=an%3A0586.46011 ).
answered Mar 21, 2020 at 16:45
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$\begingroup$ Dear Prof. Werner, thank you very much for such a quick reply and pointing me to such an interesting paper! $\endgroup$ Mar 22, 2020 at 9:16