Robinson forcing and the quasidiagonality problem

TL;DR

This paper introduces weakened versions of key open problems in C*-algebra classification, showing their interrelations and implications for quasidiagonality and MF problems.

Contribution

It proposes new weakened problems that connect to the original open questions and provides criteria linking quasidiagonality, UCT, and MF properties.

Findings

Weakening of quasidiagonality and UCT problems introduced.

Positive solutions to weakened problems imply solutions to original problems.

Provides local criteria for the MF problem.

Abstract

We introduce weakenings of two of the more prominent open problems in the classification of $\mathrm{C}^*$-algebras, namely the quasidiagonality problem and the UCT problem. We show that the a positive solution of the conjunction of the two weaker problems implies a positive solution of the original quasidiagonality problem as well as allows us to give a local, finitary criteria for the MF problem, which asks whether every stably finite $\mathrm{C}^*$-algebra is MF.