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A C*-algebraic Hoffman-Wielandt theorem

  • Bhishan Jacelon
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Abstract

We observe that the $2$-norm distance dU,2 between the unitary orbits of normal elements in a II1 factor ℳ is equal to the $2$-Wasserstein distance between the spectral measures induced by the trace τM. Using classification and optimal transport theory, we deduce an analogous $2$-norm equation for normal operators x and y in simple, separable, unital, nuclear, 𝒵-stable C*-algebras that are either monotracial, or real rank zero with finitely many extremal traces, provided that σ(x)=σ(y) is convex. Consequently, dU,2 equips the set of approximate unitary equivalence classes of contractive normal elements of ℳ with the structure of a compact length space. The same is true of the set of equivalence classes of embeddings into the Jiang-Su algebra 𝒵 of classifiable tracial $2$-Wasserstein spaces over compact, convex planar domains.

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Cite

BibTeXRIS
BibTeX
@misc{jacelon2026calgebraic,
  title = {{A $\mathrmC^*$-algebraic Hoffman-Wielandt theorem}},
  author = {Bhishan Jacelon},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.22585},
  eprint = {2605.22585},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.22585},
}
RIS
TY  - UNPB
TI  - A C^*-algebraic Hoffman-Wielandt theorem
AU  - Bhishan Jacelon
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2605.22585
UR  - https://arxiv.org/abs/2605.22585
AB  - We observe that the $2$-norm distance d_U,2 between the unitary orbits of normal elements in a II_1 factor ℳ is equal to the $2$-Wasserstein distance between the spectral measures induced by the trace τ_M. Using classification and optimal transport theory, we deduce an analogous $2$-norm equation for normal operators x and y in simple, separable, unital, nuclear, 𝒵-stable C^*-algebras that are either monotracial, or real rank zero with finitely many extremal traces, provided that σ(x)=σ(y) is convex. Consequently, d_U,2 equips the set of approximate unitary equivalence classes of contractive normal elements of ℳ with the structure of a compact length space. The same is true of the set of equivalence classes of embeddings into the Jiang-Su algebra 𝒵 of classifiable tracial $2$-Wasserstein spaces over compact, convex planar domains.
ER  -