Dynamical system problem studied by Bost--Connes

We state the problem solved by Bost and Connes in [BC] (J-B. Bost, A. Connes, Selecta Math. (New Series), 1, (1995) 411-457) and its analogue for number fields, considered in [HL](D.  Harari, E. Leichtnam, Selecta Mathematica, New Series 3 (1997), 205-243), [ALR](J. Arledge, M. Laca, I. Raeburn, Doc. Mathematica 2 (1997) 115-138) and [Coh](Paula B Cohen, J. Théorie des Nombres de Bordeaux, 11 (1999), 15-30). A (unital) $C^*$-algebra $B$ is an (unital) algebra over the complex numbers ${\mathbb C}$ with an adjoint $x\mapsto x^*$, $x\in B$, that is, an anti-linear map with $x^{**}=x$, $(xy)^*=y^*x^*$, $x,y\in B$, and a norm $\Vert\,.\,\Vert$ with respect to which $B$ is complete and addition and multiplication are continuous operations. One requires in addition that $\Vert xx^*\Vert=\Vert x\Vert^2$ for all $x\in B$. A $C^\ast$ dynamical system is a pair $(B,\sigma_t)$, where $\sigma_t$ is a 1-parameter group of $C^*$-automorphisms $\sigma:{\mathbb R}\mapsto{\rm Aut}(B)$. A state $\varphi$ on a $C^*$-algebra $B$ is a positive linear functional on $B$ satisfying $\varphi(1)=1$. The definition of Kubo-Martin-Schwinger (KMS) of an equilibrium state at inverse temperature $\beta$ is as follows.

Definition: Let $(B,\sigma_t)$ be a dynamical system, and $\varphi$ a state on $B$. Then $\varphi$ is an equilibrium state at inverse temperature $\beta$, or ${\rm KMS}_\beta$-state, if for each $x,y\in B$ there is a function $F_{x,y}(z)$, bounded and holomorphic in the band $0<{\rm Im}(z)<\beta$ and continuous on its closure, such that for all $t\in {\mathbb R}$,

$$F_{x,y}(t)=\varphi(x\sigma_t(y)),\qquad F_{x,y}(t+\sqrt{-1}\beta)=\varphi(\sigma_t(y)x).$$ (1)

A symmetry group $G$ of the dynamical system $(B,\sigma_t)$ is a subgroup of ${\rm Aut}(B)$ commuting with $\sigma$:

$$g\circ\sigma_t=\sigma_t\circ g,\qquad g\in G, t\in {\mathbb R}.$$

Consider now a system $(B,\sigma_t)$ with interaction. Then, guided by quantum statistical mechanics, we expect that at a critical temperature $\beta_0$ a phase transition occurs and the symmetry is broken. The symmetry group $G$ then permutes transitively a family of extremal ${\rm KMS}_\beta$- states generating the possible states of the system after phase transition: the ${\rm KMS}_\beta$-state is no longer unique. This phase transition phenomenon is known as spontaneous symmetry breaking at the critical inverse temperature $\beta_0$. We state the problem related to the Riemann zeta function and solved by Bost and Connes.

Problem 1: Construct a dynamical system $(B,\sigma_t)$ with partition function the zeta function $\zeta(\beta)$ of Riemann, where $\beta>0$ is the inverse temperature, having spontaneous symmetry breaking at the pole $\beta=1$ of the zeta function with respect to a natural symmetry group.

The symmetry group turns out to be the unit group of the ideles, given by $W=\prod_p {\mathbb Z}_p^*$ where the product is over the primes $p$ and ${\mathbb Z}_p^*=\{u_p\in{\mathbb Q}_p: {\vert u_p\vert}_p=1\}$. We use here the normalisation $\vert p\vert _p=p^{-1}$. This is the same as the Galois group ${\rm Gal}({\mathbb Q}^{ab}/{\mathbb Q})$, where ${\mathbb Q}^{ab}$ is the maximal abelian extension of the rational number field ${\mathbb Q}$. The interaction detected in the phase transition comes about from the interaction between the primes coming from considering at once all the embeddings of the non-zero rational numbers ${\mathbb Q}^*$ into the completions ${\mathbb Q}_p$ of ${\mathbb Q}$ with respect to the prime valuations $\vert.\vert _p$. The following natural generalisation of this problem to the number field case and the Dedekind zeta function was solved in [Coh] (see also [HL], [ALR]).

Problem 2: Given a number field $K$, construct a dynamical system $(B,\sigma_t)$ with partition function the Dedekind zeta function $\zeta_K(\beta)$, where $\beta>0$ is the inverse temperature, having spontaneous symmetry breaking at the pole $\beta=1$ of the Dedekind function with respect to a natural symmetry group.

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