Local Reflexivity and Exactness for Reduced $C^*$-Algebras of Wreath Products

We prove that, for every countable non-exact discrete group $H$ and every nontrivial countable discrete group $K$, the reduced group $C^*$-algebra $C_r^*(K\wr H)$ is not locally reflexive. These provide the first known examples of non-locally reflexive reduced group $C^*$-algebras, settling a longst