Overview
This work investigates the relationship between the 2-norm distance of unitary orbits and the 2-Wasserstein distance of spectral measures for normal elements in specific C*-algebras. It focuses on $\mathrm{II}_1$ factors and a class of simple, separable, unital, nuclear, $\mathcal{Z}$-stable $\mathrm{C}^*$-algebras.
Research Context
The study builds upon concepts related to normal elements in operator algebras, specifically within $\mathrm{II}_1$ factors and $\mathrm{C}^*$-algebras. It draws on theories from classification and optimal transport to establish metrics regarding unitary equivalence. The Jiang-Su algebra $\mathcal{Z}$ is referenced in the context of equivalence classes of embeddings.
Approach
The research primarily involves two distinct settings: $\mathrm{II}_1$ factors and a class of $\mathrm{C}^*$-algebras. For $\mathrm{II}_1$ factors, the approach involves observing the relationship between the $2$-norm distance $d_{U,2}$ and the $2$-Wasserstein distance of spectral measures induced by the trace $\tau_\mathcal{M}$. For $\mathrm{C}^*$-algebras, the approach leverages classification and optimal transport theory to extend this relationship. The specific $\mathrm{C}^*$-algebras considered are those that are simple, separable, unital, nuclear, and $\mathcal{Z}$-stable, with additional conditions of being either monotracial or real rank zero with finitely many extremal traces. A further condition requires that the spectra of the normal operators, $\sigma(x)$ and $\sigma(y)$, are convex.
Findings
- 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 $\tau_\mathcal{M}$.
- An analogous $2$-norm equation holds for normal operators $x$ and $y$ in simple, separable, unital, nuclear, $\mathcal{Z}$-stable $\mathrm{C}^*$-algebras under specific conditions. These conditions are that the $\mathrm{C}^*$-algebras are either monotracial, or real rank zero with finitely many extremal traces, and that $\sigma(x)=\sigma(y)$ is convex.
- The distance $d_{U,2}$ endows the set of approximate unitary equivalence classes of contractive normal elements of a $\mathrm{II}_1$ factor $\mathcal{M}$ with the structure of a compact length space.
- 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 also possesses the structure of a compact length space under $d_{U,2}$.