Measure Convergence under Weak Domination for Unitarily Invariant Norms on Finite von Neumann Algebras

Abstract

The classical Lebesgue dominated convergence theorem (DCT) is a cornerstone in commutative measure theory, while its extension to the noncommutative setting has remained a longstanding challenge due to the barriers posed by the noncommutative structure. Let α be a unitarily invariant norm defined on a diffuse tracial von Neumann algebra (ℳ, τ), let L^α(ℳ, τ) be the completion of ℳ with respect to the norm α(which includes all L^p(ℳ, τ) as special cases). In this paper, by replacing ``almost everywhere convergence" with Nelson's convergence in measure, we establish a noncommutative analog of the dominated convergence theorem for L^α(ℳ, τ) under a { significantly weaker dominating condition} than those previously considered. For instance, in a II₁ factor ℳ, if P is a projection with τ(P)=1/n, then n² P ``dominates" every operator of norm ≤ n. As an application, we obtain a complete characterization of center-preserving *-automorphism of finite von Neumann algebras, which unifies and extends the earlier works of Kadison and Haagerup on the structure of automorphisms.

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BibTeX RIS

BibTeX

@misc{chen2026measure,
  title = {{Measure Convergence under Weak Domination for Unitarily Invariant Norms on Finite von Neumann Algebras}},
  author = {Yanni Chen and Don Hadwin and Junhao Shen and Ye Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34776},
  eprint = {2610.34776},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34776},
}

RIS

TY  - UNPB
TI  - Measure Convergence under Weak Domination for Unitarily Invariant Norms on Finite von Neumann Algebras
AU  - Yanni Chen
AU  - Don Hadwin
AU  - Junhao Shen
AU  - Ye Zhang
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.34776
UR  - https://arxiv.org/abs/2610.34776
AB  - The classical Lebesgue dominated convergence theorem (DCT) is a cornerstone in commutative measure theory, while its extension to the noncommutative setting has remained a longstanding challenge due to the barriers posed by the noncommutative structure. Let α be a unitarily invariant norm defined on a diffuse tracial von Neumann algebra (ℳ, τ), let L^α(ℳ, τ) be the completion of ℳ with respect to the norm α(which includes all L^p(ℳ, τ) as special cases). In this paper, by replacing ``almost everywhere convergence" with Nelson's convergence in measure, we establish a noncommutative analog of the dominated convergence theorem for L^α(ℳ, τ) under a { significantly weaker dominating condition} than those previously considered. For instance, in a II_1 factor ℳ, if P is a projection with τ(P)=1/n, then n^2 P ``dominates" every operator of norm ≤ n. As an application, we obtain a complete characterization of center-preserving *-automorphism of finite von Neumann algebras, which unifies and extends the earlier works of Kadison and Haagerup on the structure of automorphisms.
ER  -

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