Superreflexive tensor product spaces

TL;DR

This paper characterizes when the projective and injective tensor products of two superreflexive Banach spaces are superreflexive, showing it occurs only if one space is finite-dimensional.

Contribution

It provides a complete characterization of superreflexivity preservation under tensor products for superreflexive Banach spaces.

Findings

Superreflexivity of $X\widehat{\otimes}_\pi Y$ occurs only if one space is finite-dimensional.

Superreflexivity of $X\widehat{\otimes}_\varepsilon Y$ occurs only if one space is finite-dimensional.

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

Abstract

The aim of this note is to prove that, given two superreflexive Banach spaces $X$ and $Y$, then $X\widehat{\otimes}_\pi Y$ is superreflexive if and only if either $X$ or $Y$ is finite-dimensional. In a similar way, we prove that $X\widehat{\otimes}_\varepsilon Y$ is superreflexive if and only if either $X$ or $Y$ is finite-dimensional.