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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 1"> <mml:semantics> <mml:msub> <mml:mi/> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> factors is completely determined by tracial evaluations. We moreover use this result to solve the so-called <italic>trace problem</italic> in the case of factorial trivial <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>W</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">W^\ast</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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.