Ergodicity of the Action of the Positive Rationals on the Group of Finite Adeles and the Bost-connes Phase Transition Theorem

We study relatively invariant measures with the multiplicators Q+ 3 q 7→ q−β on the space Af of finite adeles. We prove that for β ∈ (0, 1] such measures are ergodic, and then deduce from this the uniqueness of KMSβstates for the Bost-Connes system. Combining this with a result of Blackadar and Boca-Zaharescu, we also obtain ergodicity of the action of Q∗ on the full adeles. Bost and Connes [BC] constructed a remarkable C∗-dynamical system which has a phase transition with spontaneous symmetry breaking involving an action of the Galois group Gal(Q/Q), and whose partition function is the Riemann ζ function. In their original definition the underlying algebra arises as the Hecke algebra associated with an inclusion of certain ax + b groups. Recently Laca and Raeburn [LR, L2] have realized the Bost-Connes algebra as a full corner of the crossed product algebra C0(Af )oQ+. This new look at the system has allowed to simplify significantly the proof of the existence of KMS-states for all temperatures, and the classification of KMSβ-states for β > 1 [L1]. On the other hand, for β ≤ 1 the uniqueness of KMSβ-states implies ergodicity of the action of Q+ on Af for certain measures (in particular, for the Haar measure). The aim of this note is to give a direct proof of the ergodicity, and then to show that the uniqueness of KMSβ-states easily follows from it. So let P be the set of prime numbers, Af the restricted product of the fields Qp with respect to Zp, p ∈ P , R = ∏ p Zp its maximal compact subring, W = R∗ = ∏ p Zp. The group Q+ of positive rationals is embedded diagonally into Af , and so acts by multiplication on the additive group of finite adeles. Then the Bost-Connes algebra CQ is the full corner of C0(Af ) o Q+ determined by the characteristic function of R [L2]. The dynamics σt is defined as follows [L1]: it is trivial on C0(Af ), and σt(uq) = quq, where uq is the multiplier of C0(Af ) o Q+ corresponding to q ∈ Q+. Then ([L1]) there is a one-to-one correspondence between Received by the editors November 28, 2000 and, in revised form, May 11, 2001. 1991 Mathematics Subject Classification. Primary 46L55; Secondary 28D15. This research was partially supported by Award No UM1-2092 of the Civilian Research & Development Foundation. c ©2002 American Mathematical Society