Cotype of the triple projective tensor product of Hilbert spaces

The projective tensor product of two copies of $\ell_2$ is the space of trace-class operators, which has cotype 2, a strong regularity property of Banach spaces. For three factors, other combinations of $\ell_p$ spaces are known to lose all non-trivial cotype, a fact proved through locally decodable codes from computer science. Whether $\ell_2 \otimes_\pi \ell_2 \otimes_\pi \ell_2$ keeps some finite cotype is a long-standing question of Pisier.

Problem

A Banach space $X$ has (Rademacher) cotype $q \in [2, \infty)$ if there is a constant $C$ such that for all finite sequences $x_1, \dots, x_m \in X$,

$$\Big(\sum_{i=1}^m \|x_i\|^q\Big)^{1/q} \le C\, \Big(\mathbb{E}\,\Big\|\sum_{i=1}^m \varepsilon_i x_i\Big\|^2\Big)^{1/2},$$

where $\varepsilon_1, \dots, \varepsilon_m$ are independent random signs. $X$ has non-trivial cotype if it has cotype $q$ for some $q < \infty$; by the Maurey–Pisier theorem this fails exactly when $X$ contains the spaces $\ell_\infty^n$ uniformly. The projective tensor norm on $X \otimes Y$ is

$$\|u\|_\pi = \inf\Big\{\sum_k \|x_k\|\,\|y_k\| : u = \sum_k x_k \otimes y_k\Big\},$$

and $X \otimes_\pi Y$ is the completion.

Question (Pisier). Does $\ell_2 \otimes_\pi \ell_2 \otimes_\pi \ell_2$ have non-trivial cotype?

Background

The problem was recorded by Gilles Pisier at the AIM workshop on invariants in convex geometry and Banach space theory. Tensor products are a basic source of examples in Banach space theory, and cotype governs, for instance, how random sums behave in a space. As the source recounts, the link with coding theory arose from an attempt to use Banach space methods to bound the length of locally decodable codes.

What is known

• Tomczak-Jaegermann (1974): $\ell_2 \otimes_\pi \ell_2$, the trace class, has cotype 2. Pisier later showed that $L_p \otimes_\pi L_q$ has cotype $\max\{p, q\}$ for $p, q \in [2, \infty)$ (both as recorded in the source).
• Briët, Naor and Regev (2012): if $p_1, p_2, p_3 \in (1, \infty)$ satisfy $1/p_1 + 1/p_2 + 1/p_3 \le 1$, then $\ell_{p_1} \otimes_\pi \ell_{p_2} \otimes_\pi \ell_{p_3}$ fails to have finite cotype; they also show that for every $p \in (1, \infty)$ there is a space $X$ of finite cotype such that $\ell_p \otimes_\pi X$ does not. The proof goes through locally decodable codes. The case $p_1 = p_2 = p_3 = 2$, where the sum of reciprocals is $3/2$, is not covered.
• Giladi, Prochno, Schütt, Tomczak-Jaegermann and Werner (2017) computed volume ratios of $\ell_p^n \otimes_\pi \ell_q^n \otimes_\pi \ell_r^n$ and derived information on the cotype 2 constants of these spaces.

What would count as progress

• A proof that $\ell_2 \otimes_\pi \ell_2 \otimes_\pi \ell_2$ has cotype $q$ for some finite $q$, or that it contains $\ell_\infty^n$ uniformly (for instance via new lower bounds for locally decodable codes in the relevant regime).
• Quantitative estimates: the growth in $n$ of the cotype constants of $\ell_2^n \otimes_\pi \ell_2^n \otimes_\pi \ell_2^n$. Lower bounds can be supported by explicit families of tensors, whose norms can be estimated numerically for small $n$.
• The companion question from the same workshop: does $L_p \otimes_\pi L_q$ have finite cotype for $p \in (1, 2)$ and $q \in (1, 2]$?

References

1. J. Briët, A. Naor and O. Regev, Locally decodable codes and the failure of cotype for projective tensor products, Electron. Res. Announc. Math. Sci. 19 (2012), 120–130; arXiv:1208.0539.
2. O. Giladi, J. Prochno, C. Schütt, N. Tomczak-Jaegermann and E. Werner, On the geometry of projective tensor products, J. Funct. Anal. 273 (2017), 471–495.

Source

Imported from AIM Problem Lists convexbanach 2.1, curated by American Institute of Mathematics. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.