Inner C^∗-Selfless Groups

@misc{amrutam2026inner,
  title = {{Inner $\mathrmC^\ast$-Selfless Groups}},
  author = {Tattwamasi Amrutam and Mario Klisse and Matteo Pagliero},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40198},
  eprint = {2610.40198},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40198},
}

TY  - UNPB
TI  - Inner C^∗-Selfless Groups
AU  - Tattwamasi Amrutam
AU  - Mario Klisse
AU  - Matteo Pagliero
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40198
UR  - https://arxiv.org/abs/2610.40198
AB  - We introduce the notion of inner C^∗-selfless groups and study their structural properties. We prove a Fourier coefficient extraction theorem for invariant subalgebras of crossed products by such groups under a compatibility assumption on the action. Consequently, every inner C^∗-selfless group has the relative C^∗-invariant subalgebra rigidity property, and every finite product of such groups has the C^∗-invariant subalgebra rigidity property. We also introduce another group-theoretic notion, which we call cyclic Powers property. This implies inner C^∗-selflessness, complete C^∗-selflessness, and the Powers property, among others. We establish it for groups admitting a topologically free extreme boundary action on a compact Hausdorff space with more than two points, and for Zariski-dense discrete subgroups of PSL(d,𝕂) containing a biproximal element, where 𝕂 is a local field and d≥2. In particular, we obtain the Powers property for PSL(d,ℤ), d≥3, answering a question posed more than twenty years ago by Bridson-de la Harpe. Finally, we show that every non-trivial mixed identity-free discrete group has the relative invariant subalgebra rigidity property for its group von Neumann algebra. This shows that the Baumslag-Solitar groups BS(m,n) with |m|,|n|≥2 and |m|≠|n| have the relative ISR property, thereby answering a question of Dudko-Jiang.
ER  -