Higher projective tensor products of $c_0$

TL;DR

This paper investigates the structural differences between higher and lower order projective tensor products of Banach spaces, showing that under certain conditions, these tensor products are not isomorphic to each other's subspaces or quotients.

Contribution

It establishes that for certain Banach spaces, higher order tensor products cannot be embedded into quotients of lower order tensor products, highlighting fundamental non-isomorphism properties.

Findings

Higher tensor powers of $c_0$ are not isomorphic to subspaces of quotients of lower tensor powers.

Under specific assumptions, the non-isomorphism extends to a broad class of Banach spaces.

The results clarify the structural complexity of tensor products in Banach space theory.

Abstract

Let $m,n$ be positive integers with $m<n$. Under certain assumptions on the Banach space $X$, we prove that the $n$-fold projective tensor product of $X$, $\widehat{\otimes}^n_\pi X$, is not isomorphic to any subspace of any quotient of the $m$-fold projective tensor product, $\widehat{\otimes}_\pi^m X$. In particular, we prove that $\widehat{\otimes}^n_\pi c_0$ is not isomorphic to any subspace of any quotient of $\widehat{\otimes}_\pi^m c_0$.