On injective tensor powers of $\ell_1$

TL;DR

This paper demonstrates that the 3-fold injective tensor product of  spaces is not isomorphic to any subspace of the 2-fold tensor product, solving a problem related to tensor products of classical Banach spaces.

Contribution

It provides a new solution to Diestel's problem by showing the non-isomorphism of certain 3-fold tensor products, advancing understanding of tensor product structures in Banach space theory.

Findings

3-fold injective tensor product of  is not isomorphic to any subspace of the 2-fold tensor product

The result addresses a problem of Diestel on projective tensor products of c0

Implication for tensor products of continuous functions on compact spaces

Abstract

In this paper we prove that the $3$-fold injective tensor product $\ell_1 \widehat{\otimes}_\varepsilon \ell_1 \widehat{\otimes}_\varepsilon \ell_1 $ is not isomorphic to any subspace of $\ell_1 \widehat{\otimes}_\varepsilon \ell_1$. This result provides a new solution to a problem of Diestel on the projective tensor products of $c_0.$ Moreover, this result implies that for any infinite countable compact space $K,$ the $3$-fold projective tensor product $C(K) \widehat{\otimes}_\pi C(K)\widehat{\otimes}_\pi C(K)$ is not isomorphic to any quotient of $C(K) \widehat{\otimes}_\pi C(K)$.