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PreprintCC BY 4.0arXiv

1-Bounded Entropy for C*-Algebras

  • Connor MacMahon
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Abstract

Towards a question of Voiculescu, two notions of $1$-bounded entropy, h and htop, are defined for C*-algebras. The quantity h involves the tracial completion with respect to all traces while the quantity htop depends on operator norm microstates. It is demonstrated that htop(π’œ) does not exceed h(π’œ) for all C*-algebras π’œ. Moreover, a variational principle enabling the computation of h is introduced. The quantity h is computed in various examples, and the computation is used to justify structural conclusions such as C*-primeness, crossed product indecomposability, and free indecomposability. These notions of entropy are generalized to operator systems, where it is demonstrated that both may be computed on a basis.

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Cite

BibTeXRIS
BibTeX
@misc{macmahon20261bounded,
  title = {{1-Bounded Entropy for $C^*$-Algebras}},
  author = {Connor MacMahon},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.15409},
  eprint = {2608.15409},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.15409},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - 1-Bounded Entropy for C^*-Algebras
AU  - Connor MacMahon
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.15409
UR  - https://arxiv.org/abs/2608.15409
AB  - Towards a question of Voiculescu, two notions of $1$-bounded entropy, h and h^top, are defined for C^*-algebras. The quantity h involves the tracial completion with respect to all traces while the quantity h^top depends on operator norm microstates. It is demonstrated that h^top(π’œ) does not exceed h(π’œ) for all C^*-algebras π’œ. Moreover, a variational principle enabling the computation of h is introduced. The quantity h is computed in various examples, and the computation is used to justify structural conclusions such as C^*-primeness, crossed product indecomposability, and free indecomposability. These notions of entropy are generalized to operator systems, where it is demonstrated that both may be computed on a basis.
ER  -