Extension of exact square-ideal separation to nonseparable atomless L1 spaces

Determine whether the exact separation between the Dunford–Pettis ideal and the closed square of that ideal extends to every nonseparable finite atomless L1 space.

Background

The paper proves that the Dunford–Pettis ideal on L1[0,1] has a closed square that is strictly smaller than the ideal and establishes an exact distance formula separating certain convolution operators from that square. It also transfers the obstruction to one-sided approximate identities to arbitrary finite measure spaces whose measure algebras have a nonzero atomless part by using a complemented copy of L1[0,1].

The authors explicitly state that this complemented-copy argument does not establish whether the stronger exact square-ideal separation holds on every nonseparable finite atomless L1 space. The open problem is therefore to determine whether the separation theorem, rather than only the approximate-identity obstruction, admits this full generalization.