# Asymptotically symmetric spaces with hereditarily non-unique spreading models

## TL;DR

This paper studies a special Banach space that admits exactly two spreading models in every infinite dimensional subspace, proving it is asymptotically symmetric, which answers a previously open problem.

## Contribution

It introduces a Banach space with exactly two spreading models per subspace and proves its asymptotic symmetry, addressing an open question in the field.

## Key findings

- The space admits exactly two spreading models in every infinite dimensional subspace.
- The space is asymptotically symmetric.
- Provides a negative answer to a problem posed by Junge, the first author, and Odell.

## Abstract

We examine a variant of a Banach space $\mathfrak{X}_{0,1}$ defined by Argyros, Beanland, and the second named author that has the property that it admits precisely two spreading models in every infinite dimensional subspace. We prove that this space is asymptotically symmetric and thus it provides a negative answer to a problem of Junge, the first. named author, and Odell.