Monotone completeness conjecture for AW-algebras

Determine whether every AW-algebra is monotone complete, where monotone completeness means that every norm-bounded upward-directed family in the self-adjoint part has a least upper bound in the self-adjoint order.

Background

AW-algebras are algebraic analogues of von Neumann algebras whose projection lattices are complete. Monotone completeness requires every norm-bounded upward-directed family of self-adjoint elements to possess a least upper bound in the self-adjoint order. The paper notes that the conjecture has affirmative answers for W-algebras and commutative AW-algebras, together with several partial affirmative results, but does not resolve the general case.

The paper proves that every AW-algebra is normal and establishes an equiconsistency reduction: a model of ZFC containing a non-monotone-complete AW-algebra exists if and only if a model of ZFC containing a non-monotone-complete AW-factor exists. Thus, the general monotone completeness conjecture remains an unresolved problem, reduced to its factor case at the level of consistency strength.