Let $\Gamma$ be a discrete group. When $\Gamma$ is nonamenable, the reduced and full group $C$∗-algebras differ and it is generally believed that there should be many intermediate $C$∗-algebras, however few examples are known. In this paper, we give new constructions and compare existing constructions of intermediate group $C$∗-algebras for both generic and specific groups $\Gamma$.
Matthew Wiersma. "Constructions of exotic group $C$∗-algebras." Illinois J. Math. 60 (3-4) 655 - 667, Fall and Winter 2016. https://doi.org/10.1215/ijm/1506067285