A Godefroy-Kalton principle for free Banach lattices (Q2143241)

For a Banach space \(E\), the (non-linear) canonical embedding \(\delta\) into its free space \(F(E)\) (the predual of Lip\(_0(E)\)) can be seen to have a linear left inverse, i.e., there exists a linear and bounded operator \(\beta_E:F(E)\to E\) such that \(\beta_E(\delta (x))=x\) for all \(x\in E\). For quite a big class of Banach spaces (including all separable spaces) this \(\beta_E\) has a linear and bounded right inverse, and so \(E\) linearly embeds as a subspace of its free space. This is the Godefroy-Kalton principle, first detected in the seminal paper [\textit{G. Godefroy} and \textit{N. J. Kalton}, Stud. Math. 159, No. 1, 121--141 (2003; Zbl 1059.46058)]. As a consequence, when \(E\) has this Lipschitz lifting property and embeds isometrically into the Banach space \(F\), then it is linearly and isometrically isomorphic to some subspace of \(F\). \textit{A. Avilés} et al., building on a construction due to \textit{B. de Pagter} and \textit{A. W. Wickstead} [Proc. R. Soc. Edinb., Sect. A, Math. 145, No. 1, 105--143 (2015; Zbl 1325.46020)], have demonstrated in [J. Funct. Anal. 274, No. 10, 2955--2977 (2018; Zbl 1400.46015)] how a free lattice \(FBL[E]\) can be constructed from a Banach space \(E\) and investigate in the paper under review when the following Godefroy-Kalton principle can be obtained: When does \(\beta_X:FBL[X]\to X\), where \(X\) is a Banach lattice, have a right lattice homeomorphism inverse? Banach lattices with this property are defined to have the lattice-lifting property. In Section 2 we learn that this is the case for projective Banach lattices and lattices of the type \(FBL[E]\), where \(E\) is a Banach space. The latter implies that \(X\) has the lattice-lifting property if and only if it is lattice isomorphic to a lattice-complemented sublattice of \(FBL[X]\). In Section 3 it is shown that only when \(K\) is a neighborhood retract of \(B_{C(K)^\ast}\) (in its weak-star topology) the Banach lattice \(C(K)\) (\(K\) compact Hausdorff) can have the lattice-lifting property. In the metrizable case \(K\) is an absolute neighborhood retract. This implies that for metrizable \(K\) the lattice-lifting property is equivalent to \(C(K)\) being a projective Banach lattice. It is stated as an open question whether the lattice lifting property of \(C(K)\) implies that \(K\) is an absolute neighborhood retract without any further conditions on \(K\). In the rather technical Section 4 the authors show that Banach spaces \(X\) with a 1-unconditional basis (coordinatewise order) have the lattice-lifting property, even with the right inverse of \(\beta_X\) having norm one. The arguments use ideas of the particular case of \(c_0\), which was proved earlier in [\textit{A. Avilés} et al., Rev. Mat. Complut. 34, No. 1, 203--213 (2021; Zbl 1462.46016)].