Optimal transport and unitary orbits in C⁎-algebras

Two areas of mathematics which have received substantial attention in recent years are the theory of optimal transport and the Elliott classification programme for C⁎-algebras. We combine these two seemingly unrelated disciplines to make progress on a classical problem of Weyl. In particular, we show how results from the Elliott classification programme can be used to translate continuous transport of spectral measures into optimal unitary conjugation in C⁎-algebras. As a consequence, whenever two normal elements of a sufficiently well-behaved C⁎-algebra share a spectrum amenable to such continuous transport, and have trivial K1-class, the distance between their unitary orbits can be computed tracially.