Invariant $C^*$-subalgebras of the reduced group $C^*$-algebra

Tattwamasi Amrutam; Yongle Jiang

doi:10.1016/j.jfa.2026.111471

Invariant $C^$-subalgebras of the reduced group $C^$-algebra

Operator Algebras 2026-03-26 v3 Dynamical Systems Functional Analysis

Authors: Tattwamasi Amrutam , Yongle Jiang

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Abstract

Let $\Gamma$ be a countable discrete group. We say that $\Gamma$ has $C^*$ -invariant subalgebra rigidity (ISR) property if every $\Gamma$ -invariant $C^*$ -subalgebra $\mathcal{A}\le C_r^*(\Gamma)$ is of the form $C_r^*(N)$ for some normal subgroup $N\triangleleft\Gamma$ . We show that all torsion-free, non-amenable (cylindrically) hyperbolic groups with property-AP and a finite direct product of such groups have this property. We also prove that an infinite group $\Gamma$ has the C $^*$ -ISR property only if $\Gamma$ is simple amenable or $C^*$ -simple.

Keywords

C*-algebra reduced group C*-algebra invariant subalgebra hyperbolic groups property-AP C*-simple amenable groups

Cite

@article{arxiv.2503.09548,
  title  = {Invariant $C^*$-subalgebras of the reduced group $C^*$-algebra},
  author = {Tattwamasi Amrutam and Yongle Jiang},
  journal= {arXiv preprint arXiv:2503.09548},
  year   = {2026}
}

Comments

Final version: all the referee's suggestions have been implemented