The class of Banach lattices is not primary

2025/08/28 by Acuaviva, Antonio · 1 citation
#46B03 #46B42 #46E15 #54G12 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2508.21018

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) ≃ X ⊕ X and neither X nor X is isomorphic to a Banach lattice. In particular, it also follows that the class of C(K)-spaces is not primary.

Citations

• Complemented subspaces of Banach lattices
• A negative solution to the complemented subspace problem in Banach lattices
• The complemented subspace problem for C(K)-spaces: A counterexample

Cited by

• C(K)-spaces with few operators relative to posets

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• Complemented subspaces of Banach lattices
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• The complemented subspace problem for C(K)-spaces: A counterexample
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