Lifting orthogonality to orthogonality

Determine whether orthogonality of elements or unitaries in an ultrapower of a tracial von Neumann algebra can always be lifted to orthogonality of representatives in the original algebra.

Background

The paper studies lifting finite-order independence properties from ultrapowers of tracial von Neumann algebras to the underlying algebras. Prior work established lifting results for certain orthogonal unitaries, including unitaries of finite order and, under stronger 2-independence assumptions, arbitrary unitaries.

The paper proves that 2N-independence can be lifted to N-independence, but it does not resolve the more basic general question of whether orthogonality itself lifts to orthogonality. This question is therefore explicitly identified as remaining open outside the special cases treated in earlier work.