Semiprojective Banach lattices

Abstract

We introduce a norm-controlled notion of semiprojectivity for Banach lattices, requiring liftability of contractive lattice homomorphisms through inductive limits of closed ideals with arbitrarily small loss of norm control. Our main result establishes that, for a compact metric space $X$, the Banach lattice $C(X)$ is semiprojective if and only if $X$ is an absolute neighbourhood retract. Notably, this characterisation is strictly more permissive than its $C^*$-algebraic counterpart: by a theorem of S\orensen and Thiel, $C(X)$ is semiprojective in the category of $C^*$-algebras and $*$-homomorphisms if and only if $X$ is an ANR of dimension at most one. The dimensional obstruction disappears in the Banach-lattice setting because lattice homomorphisms between $C(K)$-spaces are automatically weighted composition operators, and therefore no commutation relations need to be lifted. We also show that uncountable $\ell_1$-sums of $1^+$-projective Banach lattices with topological order units are semiprojective but need not be $1^+$-projective, establishing that the two notions are genuinely distinct. On the negative side, we prove that $\ell_p$ and $L_p([0,1])$ for $p \in (1,\infty)$ as well as Orlicz spaces are not semiprojective.