The class of Banach lattices is not primary

Acuaviva Huertos, Antonio (2026) The class of Banach lattices is not primary. Forum of Mathematics, Sigma, 14: e41. ISSN 2050-5094

Abstract

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