Bounded approximation property without bounded positive approximation property

Determine whether there exists a Banach lattice with the ordinary bounded approximation property but without the bounded positive approximation property.

Background

The bounded positive approximation property (BPAP) strengthens the ordinary bounded approximation property by requiring the approximating finite-rank operators to be positive. The paper notes that only a few Banach lattices are known to fail BPAP, and that the relationship between the ordinary bounded approximation property and BPAP remains unresolved in the relevant direction. Specifically, it is unknown whether the ordinary bounded approximation property can hold for a Banach lattice that lacks BPAP.

References

Only a few Banach lattices are known which do not satisfy the BPAP; indeed, it is not even known whether there are Banach lattices without the BPAP which have the ordinary bounded approximation property (see, e.g., p. 14).

— Complemented Copies of $c_{0}$ in Positive Tensor Products of Banach Lattices  (2608.24834 - Melnikov, 25 Aug 2026) in Section 2, subsection “Approximation Properties in Banach Lattices”