Continuous selection of unitaries in II₁ factors

We prove continuous-valued analogues of the basic fact that Murray–von Neumann subequivalence of projections in II 1 _1 factors is completely determined by tracial evaluations. We moreover use this result to solve the so-called trace problem in the case of factorial trivial W ∗ W^ -bundles whose base space has covering dimension at most 1. Our arguments are based on applications to von Neumann algebras of a continuous selection theorem due to Michael.