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PreprintCC BY 4.0arXiv

On the Isomorphism of Semicircular Algebras

  • D. Shlyakhtenko
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Abstract

Building on OpenAI's proof of isomorphism of free group factors, we give a self-contained proof of a corresponding result for C*-algebras generated by free semicircular systems: the isomorphism class of a C*-algebra generated by a semicircular n-tuple does not depend on n∈{2,3,…,+∞}. Similarly, reduced free products of n copies of the Jiang--Su algebra are isomorphic for different n in this range. While reduced free group C*-algebras of different ranks are not isomorphic (by a result of Pimsner--Voiculescu), we show that they have strong free absorption properties. At the heart of the proof is an automorphism approximation property for inclusions of C*- and W*-algebras; a relative version of that property was previously used to derive free complementation results.

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Cite

BibTeXRIS
BibTeX
@misc{shlyakhtenko2026isomorphism,
  title = {{On the Isomorphism of Semicircular Algebras}},
  author = {D. Shlyakhtenko},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19590},
  eprint = {2610.19590},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19590},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the Isomorphism of Semicircular Algebras
AU  - D. Shlyakhtenko
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.19590
UR  - https://arxiv.org/abs/2610.19590
AB  - Building on OpenAI's proof of isomorphism of free group factors, we give a self-contained proof of a corresponding result for C^*-algebras generated by free semicircular systems: the isomorphism class of a C^*-algebra generated by a semicircular n-tuple does not depend on n∈{2,3,…,+∞}. Similarly, reduced free products of n copies of the Jiang--Su algebra are isomorphic for different n in this range. While reduced free group C^*-algebras of different ranks are not isomorphic (by a result of Pimsner--Voiculescu), we show that they have strong free absorption properties. At the heart of the proof is an automorphism approximation property for inclusions of C^*- and W^*-algebras; a relative version of that property was previously used to derive free complementation results.
ER  -