# On the uniform Roe algebra as a Banach algebra and embeddings of $\ell_p$ uniform Roe algebras

## TL;DR

This paper explores the structure and embeddings of $ell_p$ uniform Roe algebras, extending previous results and establishing rigidity properties based solely on their Banach algebra structure.

## Contribution

It generalizes mutual embedding results for $ell_p$ uniform Roe algebras and proves rigidity results for classical uniform Roe $ ext{C}^*$-algebras.

## Key findings

- Mutual embeddings of $ell_p$ uniform Roe algebras are characterized.
- Rigidity results show the algebraic structure determines the algebra uniquely.
- Extensions of previous results to a broader class of operator algebras.

## Abstract

We work on $\ell_p$ uniform Roe algebras associated to metric spaces, and on their mutual embedding. We generalize results of I. Farah and the authors to mutual embeddings of uniform Roe algebras of operators on $\ell_p$ spaces. Simultaneously, we obtain rigidity results for the classic uniform Roe $\mathrm{C}^*$-algebras which depend only on their Banach algebra structure.