The class of Banach lattices is not primary

Abstract Building on a recent construction of Plebanek and Salguero-Alarcón, which solved the Complemented Subspace Problem for $C(K)$ -spaces, and the subsequent work of De Hevia, Martínez-Cervantes, Salguero-Alarcón, 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.