Continuous Selection of Unitaries in II$_1$ Factors

TL;DR

This paper develops continuous analogues of subequivalence of projections in II$_1$ factors and applies these results to solve the trace problem for certain trivial $W^*$-bundles with low-dimensional base spaces, using a continuous selection theorem.

Contribution

It introduces continuous-valued analogues of projection subequivalence in II$_1$ factors and applies them to the trace problem in trivial $W^*$-bundles with low-dimensional bases.

Findings

Established continuous analogues of Murray-von Neumann subequivalence.

Solved the trace problem for factorial trivial $W^*$-bundles over spaces with dimension ≤ 1.

Applied a continuous selection theorem to von Neumann algebras.

Abstract

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.