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 longstanding existence question. Moreover, for every nontrivial countable exact discrete group $K$, we establish that local reflexivity of $C_r^*(K\wr H)$ is equivalent to exactness of $H$, and hence to exactness of $C_r^*(K\wr H)$. This gives a partial affirmative answer to Kirchberg's question of whether local reflexivity implies exactness for $C^*$-algebras. As applications, we construct finitely generated groups with the Haagerup property, as well as finitely generated residually finite groups, whose reduced group $C^*$-algebras are not locally reflexive. In particular, neither the Haagerup property nor coarse embeddability into Hilbert space guarantees local reflexivity of the associated reduced group $C^*$-algebra.