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

@misc{dong2026local,
  title = {{Local Reflexivity and Exactness for Reduced $C^*$-Algebras of Wreath Products}},
  author = {Pingchuan Dong and Zhe Dong},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34695},
  eprint = {2610.34695},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34695},
}

TY  - UNPB
TI  - Local Reflexivity and Exactness for Reduced C^*-Algebras of Wreath Products
AU  - Pingchuan Dong
AU  - Zhe Dong
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.34695
UR  - https://arxiv.org/abs/2610.34695
AB  - 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^*(Kwr 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^*(Kwr H) is equivalent to exactness of H, and hence to exactness of C_r^*(Kwr 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.
ER  -