The positive approximation property of Banach lattices (Q1116407)

A Banach lattice X is said to have the positive approximation property (p.a.p.) if the identity can be approximated uniformly on compact sets by positive, bounded finite rank operators. It is an open problem whether every Banach lattice with the usual approximation property also has the p.a.p. The author gives several conditions on a Banach lattice X, equivalent to X having bounded p.a.p. They look to some extent similar to the Banach spaces setting but the proofs are essentially different. The cause is in the fact that perturbations of operators often spoil the positivity. The approach to positive perturbations, developed in this paper, has its own interest.