# The class of Banach lattices is not primary

## TL;DR

This paper demonstrates that the class of Banach lattices is not primary by constructing a specific example of a compact space where the associated function space decomposes into parts that are not Banach lattices.

## Contribution

It provides the first explicit example showing the class of Banach lattices is not primary, extending recent solutions to the Complemented Subspace Problem.

## Key findings

- Constructed a compact space L with C(L) decomposing into non-Banach lattice parts.
- Showed that the class of C(K)-spaces is not primary.
- Extended the understanding of the structure of Banach lattices and their complemented subspaces.

## Abstract

Building on a recent construction of Plebanek and Salguero-Alarc\'on, which solved the Complemented Subspace Problem for $C(K)$-spaces, and the subsequent work of De Hevia, Mart\'inez-Cervantes, Salguero-Alarc\'on, and Tradacete solving the Complemented Subspace Problem for Banach lattices, we show that the class of Banach lattices is not primary. Specifically, we exhibit a compact Hausdorff space $L$ such that $C(L) \simeq X \oplus \tilde{X}$ and neither $X$ nor $\tilde{X}$ is isomorphic to a Banach lattice. In particular, it also follows that the class of $C(K)$-spaces is not primary.