A $\mathrm{C}^*$-algebraic Hoffman-Wielandt theorem

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 $\tau_\mathcal{M}$. 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 $\sigma(x)=\sigma(y)$ 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.