It is shown that for every $p\in (1,\infty)$ there exists a Banach space $X$ of finite cotype such that the projective tensor product $\ell_p\tp X$ fails to have finite cotype. More generally, if $p_1,p_2,p_3\in (1,\infty)$ satisfy $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\le 1$ then $\ell_{p_1}\tp\ell_{p_2}\tp\ell_{p_3}$ does not have finite cotype. This is proved via a connection to the theory of locally decodable codes.

Additional Metadata
Keywords cotype, projective tensor product, locally decodable codes
THEME Life Sciences (theme 5), Logistics (theme 3)
Publisher American Institute of Mathematical Sciences
Journal Electronic Research Announcements in Mathematical Sciences
Project Qubit Applications
Briët, J, Naor, A, & Regev, O. (2012). Locally decodable codes and the failure of cotype for projective tensor products. Electronic Research Announcements in Mathematical Sciences, 19, 120–130.