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 $τ_\mathcal{M}$. Using classification and optimal transport theory, we d