Approximation of definable functions by C^1 definable functions in the non-commutative setting

Determine whether definable functions (in the language of tracial von Neumann algebras) can be approximated by an appropriate class of C^1 definable functions, possessing derivatives suitable for applying change-of-variables arguments (e.g., Jacobian determinants) in non-commutative optimal transport.

Background

The need for differentiable approximations arises in attempts to prove geodesic concavity of χ_full by computing entropy along transport maps and evaluating Jacobians. The available transport maps are bi-Lipschitz but may lack well-defined derivatives.

If definable functions could be approximated by a C^1 class (in a sense compatible with the non-commutative framework), one could potentially apply analytic tools similar to the classical setting to advance results on entropy along geodesics.