Non-MF groups and non-finite full group C*-algebras

If $\Gamma$ is a non-coHopfian property-(T) group and $G$ is its ascending HNN extension, then the full group algebra $C^*(G)$ is not finite. Indeed, Kazhdan projections $p_H<p_\Gamma$ become unitarily equivalent under the stable letter, which cannot occur inside a finite $C^*$-algebra. For $$W=\left(\bigoplus_{G/\Gamma}\mathbb Z/2\mathbb Z\right)\rtimes G,$$ the paper proves more strongly that every homomorphism $$W\to U\!\left(\prod M_{d_n}/\bigoplus M_{d_n}\right)$$ kills an explicit nonidentity element $b_\gamma$. Hence $W$ is not MF. Consequently $C_r^*(W)$ is an explicit stably finite but non-MF $C^*$-algebra.

If $\Gamma$ is a non-coHopfian property-(T) group and $G$ is its ascending HNN extension, then the full group algebra $C^*(G)$ is not finite. Indeed, Kazhdan projections $p_H<p_\Gamma$ become unitarily equivalent under the stable letter, which cannot occur inside a finite $C^*$-algebra. For $$W=\left(\bigoplus_{G/\Gamma}\mathbb Z/2\mathbb Z\right)\rtimes G,$$ the paper proves more strongly that every homomorphism $$W\to U\!\left(\prod M_{d_n}/\bigoplus M_{d_n}\right)$$ kills an explicit nonidentity element $b_\gamma$. Hence $W$ is not MF. Consequently $C_r^*(W)$ is an explicit stably finite but non-MF $C^*$-algebra.