Complemented subspace problem for Banach spaces with unconditional bases

The problem asks whether every complemented subspace of a Banach space with an unconditional basis must itself have an unconditional basis. It was explicitly recorded as open in earlier literature.

Known results

• Casazza and Kalton (1996) constructed a space whose complemented subspaces need not retain uniqueness of an unconditional basis, but did not show that they lack every unconditional basis.
• A 2007 paper resolved a related quotient-embedding problem for reflexive spaces while stating that the complemented-subspace question remained open.

September 2026 claimed counterexample

A new preprint claims negative solutions by constructing superreflexive spaces with complemented subspaces, and duals, lacking unconditional bases; it therefore claims to settle the problem and related questions. The claim is unverified.

Current status (as of September 2026): A preprint claims the problem is solved negatively, but independent mathematical verification remains outstanding.