Relative norm averaging property and selflessness of almost periodic actions

Abstract

Let D be a unital separable C^*-algebra with a faithful state σ and G be a discrete group. We introduce a property called relative norm averaging property (RNA) for almost periodic σ-preserving actions by G on (D,σ). We prove that if an almost periodic σ-preserving action α satisfies that α(G)̅ ⊆ Aut(D,σ) is connected and nontrivial, then α satisfies (RNA). As an application, we prove that if G has property PHP of Ozawa and α:Gcurvearrowright (D,σ) is an ergodic (i.e. D^G=ℂ) almost periodic σ-preserving action with α(G)̅ being connected, then Drtimes_r G is selfless. This includes certain paradoxical isometric actions of 𝔽₂ on the sphere. In non-ergodic case, we prove that the relative norm averaging property implies that Drtimes_r G satisfies the D^G-valued selfless, if G satisfies PHP.

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BibTeX RIS

BibTeX

@misc{toyota2026relative,
  title = {{Relative norm averaging property and selflessness of almost periodic actions}},
  author = {Ryo Toyota},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35644},
  eprint = {2610.35644},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35644},
}

RIS

TY  - UNPB
TI  - Relative norm averaging property and selflessness of almost periodic actions
AU  - Ryo Toyota
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35644
UR  - https://arxiv.org/abs/2610.35644
AB  - Let D be a unital separable C^*-algebra with a faithful state σ and G be a discrete group. We introduce a property called relative norm averaging property (RNA) for almost periodic σ-preserving actions by G on (D,σ). We prove that if an almost periodic σ-preserving action α satisfies that α(G)̅ ⊆ Aut(D,σ) is connected and nontrivial, then α satisfies (RNA). As an application, we prove that if G has property PHP of Ozawa and α:Gcurvearrowright (D,σ) is an ergodic (i.e. D^G=ℂ) almost periodic σ-preserving action with α(G)̅ being connected, then Drtimes_r G is selfless. This includes certain paradoxical isometric actions of 𝔽_2 on the sphere. In non-ergodic case, we prove that the relative norm averaging property implies that Drtimes_r G satisfies the D^G-valued selfless, if G satisfies PHP.
ER  -

Abstract

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