Attainment of the decomposable-norm infimum

Determine whether the infimum in the definition of the decomposable norm is always attained for linear maps between operator systems.

Background

For a linear map between operator systems, the paper defines the decomposable norm through an infimum over pairs of completely positive diagonal maps making an associated 2-by-2 matrix-valued map completely positive. The authors note that attainment is known when the range is a dual space, by weak-star compactness, but leave the general case unresolved.

References

Incidentally it remains an open question (already raised by Haagerup in ) whether the inf in the preceding definition is always attained (of course it is so, by a weak* compactness argument, when the range is a dual space).

— Operator space theory from the maximal viewpoint  (2610.21929 - Pisier et al., 15 Oct 2026) in Section 3, immediately after Remark 3.4 (before Remark \ref{nor})