Optimal transport and unitary orbits in C*-algebras

TL;DR

This paper bridges optimal transport and C*-algebra theory to compute distances between unitary orbits of normal elements via spectral measure transport, advancing the classification of C*-algebras.

Contribution

It introduces a novel approach connecting spectral measure transport with optimal unitary conjugation in C*-algebras using classification results.

Findings

Distance between unitary orbits can be computed tracially.

Continuous transport of spectral measures relates to unitary conjugation.

Applicable to normal elements with trivial K_1-class in well-behaved C*-algebras.

Abstract

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 K_1-class, the distance between their unitary orbits can be computed tracially.