Continuous Selection of Unitaries in II${}_1$ Factors

We prove continuous-valued analogues of the basic fact that Murray-von Neumann subequivalence of projections in II${}_1$ 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 $W^{\ast }$-bundles whose base space has covering dimension at most 1. Our arguments are based on applications of a continuous selection theorem due to Michael to von Neumann algebras.