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A boundary approach to rigidity of q-Araki--Woods factors

  • Yusuke Isono
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Abstract

In this paper, we study structural and rigidity properties of q-Araki--Woods factors arising from strongly continuous orthogonal representations of R. We first introduce an AO-type boundary property that is formulated in terms of length projections on the q-Fock spaces, and prove that all q-Araki--Woods factors satisfy this condition. We next characterize uniform convergence of the associated malleable deformations. We then use these two ingredients as a substitute for weak containment in the coarse bimodule and deduce several rigidity theorems in the framework of Popa's deformation/rigidity theory. As a main consequence, we prove that every q-Araki--Woods factor is stably strongly solid. We also prove a normalizer rigidity theorem for inclusions arising from direct sum representations. Finally, we compute Connes' τ-invariant, and construct a continuum of pairwise non-isomorphic type III1 q-Araki--Woods factors with the same τ-invariant.

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Cite

BibTeXRIS
BibTeX
@misc{isono2026boundary,
  title = {{A boundary approach to rigidity of $q$-Araki--Woods factors}},
  author = {Yusuke Isono},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27520},
  eprint = {2610.27520},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27520},
}
RIS
TY  - UNPB
TI  - A boundary approach to rigidity of q-Araki--Woods factors
AU  - Yusuke Isono
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.27520
UR  - https://arxiv.org/abs/2610.27520
AB  - In this paper, we study structural and rigidity properties of q-Araki--Woods factors arising from strongly continuous orthogonal representations of R. We first introduce an AO-type boundary property that is formulated in terms of length projections on the q-Fock spaces, and prove that all q-Araki--Woods factors satisfy this condition. We next characterize uniform convergence of the associated malleable deformations. We then use these two ingredients as a substitute for weak containment in the coarse bimodule and deduce several rigidity theorems in the framework of Popa's deformation/rigidity theory. As a main consequence, we prove that every q-Araki--Woods factor is stably strongly solid. We also prove a normalizer rigidity theorem for inclusions arising from direct sum representations. Finally, we compute Connes' τ-invariant, and construct a continuum of pairwise non-isomorphic type III_1 q-Araki--Woods factors with the same τ-invariant.
ER  -