Non-amenable tight squeezes by Kirchberg algebras

Abstract

We give a framework to produce C * -algebra inclusions with extreme properties. This gives the first constructive nuclear minimal ambient C * -algebras. We further obtain a purely infinite analogue of Dadarlat's modeling theorem on AF-algebras: Every Kirchberg algebra is rigidly and KK-equivalently sandwiched by non-nuclear C * -algebras without intermediate C * -algebras. Finally we reveal a novel property of Kirchberg algebras: They embed into arbitrarily wild C * -algebras as rigid maximal C * -subalgebras.