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PreprintCC BY 4.0arXiv

Sharp quantum logarithmic Sobolev inequalities via semigroup interpolation

  • Gan Yao
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Abstract

We develop a semigroup interpolation method for sharp logarithmic Sobolev inequalities using spectral calculus and estimates of Re⟨ Lx,x2⟩2, where L generates an auxiliary symmetric quantum Markov semigroup. We apply the method to free groups and to crossed products of a finite von Neumann algebra 𝒩 by ℤ or ℤn, for n=2 or n≥4, when the action commutes with the semigroup on 𝒩. In particular, the word-length generators on quantum tori and the Weyl word-length generators on Mn(ℂ) satisfy logarithmic Sobolev inequalities with optimal constant $2$.

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Cite

BibTeXRIS
BibTeX
@misc{yao2026sharp,
  title = {{Sharp quantum logarithmic Sobolev inequalities via semigroup interpolation}},
  author = {Gan Yao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25180},
  eprint = {2610.25180},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25180},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Sharp quantum logarithmic Sobolev inequalities via semigroup interpolation
AU  - Gan Yao
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25180
UR  - https://arxiv.org/abs/2610.25180
AB  - We develop a semigroup interpolation method for sharp logarithmic Sobolev inequalities using spectral calculus and estimates of Re⟨ Lx,x^2⟩_2, where L generates an auxiliary symmetric quantum Markov semigroup. We apply the method to free groups and to crossed products of a finite von Neumann algebra 𝒩 by ℤ or ℤ_n, for n=2 or n≥4, when the action commutes with the semigroup on 𝒩. In particular, the word-length generators on quantum tori and the Weyl word-length generators on M_n(ℂ) satisfy logarithmic Sobolev inequalities with optimal constant $2$.
ER  -