Trace dynamics and the complexity of cocycle conjugacy

For every nonempty compact metrizable space X, we construct a simple separable unital nuclear stably finite C*-algebra A whose extremal traces form a copy of X. On the automorphisms of A with the Rokhlin property, cocycle conjugacy is Borel bireducible with conjugacy of homeomorphisms of X. The reduction to homeomorphisms is the induced action on extremal traces, and the reverse reduction is a Borel choice of Rokhlin lifts. Taking X to be the Hilbert cube gives cocycle conjugacy the universal complexity of Polish group orbit equivalence relations. In particular, it is not classifiable by countable structures. This answers a question raised by Gardella and Lupini in 2014. The construction applies Lin’s classification of Rokhlin automorphisms to algebras with rational K₀, trivial K₁, and constant tracial pairing. GitHub repository: https://github.com/eulogica/trace-dynamics-cocycle-conjugacy