Stability and Testability

Some years ago, I proved with Shulman and Sørensen that precisely 12 of the 17 wallpaper groups are matricially stable in the operator norm. We did so as part of a general investigation of when group $C^*$-algebras have the semiprojectivity and weak matricial semiprojectivity properties — notions which are standard tools in the classification theory for $C^*$-algebras.

Our results were largely negative, and recently Dadarlat has provided a framework for understanding the obstructions to matricial stability for discrete groups. With this perspec­tive, our results may be seen as showing that, at least in this case, stability ensues when the obstructions allow it.

I intend to go through the proof of this positive result in a form aimed at non-$C^*$-algebraists. It must be admitted that the proof is very $C^*$-algebraic in nature, but it goes through a natural dimension reduction technique (invented by Friis and Rørdam in the mid-90’s) which I can definitely explain and expect could be useful in other settings as well.