A $\mathrm{C}^*$-algebraic Hoffman-Wielandt theorem
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910246349307904 |
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| author | Jacelon, Bhishan |
| author_facet | Jacelon, Bhishan |
| contents | We observe that the $2$-norm distance $d_{U,2}$ between the unitary orbits of normal elements in a $\mathrm{II}_1$ factor $\mathcal{M}$ is equal to the $2$-Wasserstein distance between the spectral measures induced by the trace $τ_\mathcal{M}$. Using classification and optimal transport theory, we deduce an analogous $2$-norm equation for normal operators $x$ and $y$ in simple, separable, unital, nuclear, $\mathcal{Z}$-stable $\mathrm{C}^*$-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 $\mathcal{M}$ 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 $\mathcal{Z}$ of classifiable tracial $2$-Wasserstein spaces over compact, convex planar domains. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_22585 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A $\mathrm{C}^*$-algebraic Hoffman-Wielandt theorem Jacelon, Bhishan Operator Algebras Metric Geometry 46L05, 46L10, 49Q22 We observe that the $2$-norm distance $d_{U,2}$ between the unitary orbits of normal elements in a $\mathrm{II}_1$ factor $\mathcal{M}$ is equal to the $2$-Wasserstein distance between the spectral measures induced by the trace $τ_\mathcal{M}$. Using classification and optimal transport theory, we deduce an analogous $2$-norm equation for normal operators $x$ and $y$ in simple, separable, unital, nuclear, $\mathcal{Z}$-stable $\mathrm{C}^*$-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 $\mathcal{M}$ 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 $\mathcal{Z}$ of classifiable tracial $2$-Wasserstein spaces over compact, convex planar domains. |
| title | A $\mathrm{C}^*$-algebraic Hoffman-Wielandt theorem |
| topic | Operator Algebras Metric Geometry 46L05, 46L10, 49Q22 |
| url | https://arxiv.org/abs/2605.22585 |