Non-abelian Bost-Connes groupoids: two no-go theorems, and a construction that realizes Prob(G/N_beta) exactly when the Frobenius lifts mix

Abstract

Two routes to a non-abelian Bost-Connes system are closed: KMS states of B x J_S are traces on B, so noncommutative coefficient algebras are invisible; and isotropy produces the dual data Irr(N) with a symmetry-fixed point, so it can never yield Prob(G/N) with a transitive action. A third route works but not canonically: ordering the primes by norm gives a Frobenius cocycle with genuinely non-commuting increments on a principal etale groupoid, whose KMS simplex is Prob(H\G) for the Mackey range H, with a non-abelian Kakutani theorem (rho o c is a coboundary iff Xi_beta(rho,S) is finite, by a matrix martingale and a determinant argument) showing N_beta is the normal closure of H. For S_3 with transposition classes the simplex depends on the lift; when N_beta is rigid, e.g. A_3, it is Prob(G/N_beta) for every lift.