With a global function field with constant field , a finite set of primes in and an abelian extension of , finite or infinite, we associate a C*-dynamical system. The systems, or at least their underlying groupoids, defined earlier by Jacob using the ideal action on Drinfeld modules and by Consani–Marcolli using commensurability of -lattices are isomorphic to particular cases of our construction. We prove a phase transition theorem for our systems and show that the unique KMS-state for every gives rise to an ITPFI-factor (ITPFI stands for “infinite tensor product of finite type I factors”) of type III, where is the degree of the algebraic closure of in . Therefore for we get a factor of type III. Its flow of weights is a scaled suspension flow of the translation by the Frobenius element on Gal.
Cite this article
Sergey Neshveyev, Simen Rustad, Bost–Connes systems associated with function fields. J. Noncommut. Geom. 8 (2014), no. 1, pp. 275–301DOI 10.4171/JNCG/156