Semiprojectivity and norm-controlled lifting in Banach lattices

We characterize semiprojectivity of a real Banach lattice by almost contractive lifting through ideal quotients of separable Banach lattices, with no separability assumption on the domain. Semiprojectivity is equivalent to ℓ-openness, implies ℓ-closedness, and is unchanged if the finite ideal stage is allowed to depend on the norm tolerance. For separable domains it coincides with 1⁺-projectivity. We prove that arbitrary ℓ₁-sums preserve semiprojectivity without an order-unit hypothesis on the summands. In contrast, ℓ₁(ℕ) ⊕∞ ℝ is not semiprojective, although both summands are 1⁺-projective. Thus semiprojectivity need not pass to finite ℓ∞-sums or split extensions. The example is ∞-projective with infimum of uniform projectivity constants equal to 2, admits an equivalent 1⁺-projective lattice norm, and is not ℓ-closed.

GitHub repository: https://github.com/eulogica/semiprojectivity-banach-lattices