# Daugavet property in tensor product spaces

## TL;DR

This paper investigates the Daugavet property in tensor product spaces of Banach spaces, establishing conditions under which the property holds for various tensor products, including those involving L1 spaces and L1-preduals.

## Contribution

It provides new results on the Daugavet property in tensor products of Banach spaces, especially for L1 spaces, L1-preduals, and symmetric tensor products, using novel techniques.

## Key findings

- L1 tensor products with non-atomic measures have the Daugavet property.
- Tensor products of L1-preduals with the Daugavet property also have it.
- Results imply consequences for roughness and symmetric tensor products.

## Abstract

We study the Daugavet property in tensor products of Banach spaces. We show that $L_1(\mu)\widehat{\otimes}_\varepsilon L_1(\nu)$ has the Daugavet property when $\mu$ and $\nu$ are purely non-atomic measures. Also, we show that $X\widehat{\otimes}_\pi Y$ has the Daugavet property provided $X$ and $Y$ are $L_1$-preduals with the Daugavet property, in particular spaces of continuous functions with this property. With the same tecniques, we also obtain consequences about roughness in projective tensor products as well as the Daugavet property of projective symmetric tensor products.