The C*-algebra of Bost--Connes

We give a different construction of the $C^*$-algebra of Bost-Connes to that found in their original paper. It is directly inspired by work of Arledge-Laca-Raeburn. Let $A_f$ denote the ring of finite adeles of ${\mathbb Q}$, that is the restricted product of ${\mathbb Q}_p$ with respect to ${\mathbb Z}_p$ as $p$ ranges over the finite primes. Recall that this restricted product consists of the infinite vectors $(a_p)_p$, indexed by the primes $p$, such that $a_p\in {\mathbb Q}_p$ with $a_p\in{\mathbb Z}_p$ for almost all primes $p$. The group of (finite) ideles ${\mathcal J}$ consists of the invertible elements of the adeles. Let ${\mathbb Z}_p^*$ be those elements of $u_p\in {\mathbb Z}_p$ with $\vert u_p\vert _p=1$. Notice that an idele $(u_p)_p$ has $u_p\in {\mathbb Q}^*_p$ with $u_p\in{\mathbb Z}_p^*$ for almost all primes $p$. Let

$${\mathcal R}=\prod_p{\mathbb Z}_p,\qquad I={\mathcal J}\cap{\mathcal R},\qquad W=\prod_p{\mathbb Z}_p^*.$$

Further, let $\bold I$ denoted the semigroup of integral ideals of ${\mathbb Z}$, which are of the form $m{\mathbb Z}$ where $m\in{\mathbb Z}$. Notice that $I$ as above is also a semigroup. We have a natural short exact sequence,

$$1\rightarrow W\rightarrow I\rightarrow{\bold I}\rightarrow 1.$$ (1)

The map $I\rightarrow{\bold I}$ in this short exact sequence is given as follows. To $(u_p)_p\in I$ associate the ideal $\prod_pp^{{\rm ord}_p(u_p)}$ where ${\rm ord}_p(u_p)$ is determined by the formula $\vert u_p\vert _p=p^{-{\rm ord}_p(u_p)}$. By the Strong Approximation Theorem we have

$${\mathbb Q}/{\mathbb Z}\simeq A_f/{\mathcal R}\simeq\oplus_p{\mathbb Q}_p/{\mathbb Z}_p$$ (2)

and we have therefore a natural action of $I$ on ${\mathbb Q}/{\mathbb Z}$ by multiplication in $A_f/{\mathcal R}$ and transport of structure. We have the following straightforward Lemmata

Lemma 1. For $a=(a_p)_p\in I$ and $y\in A_f/{\mathcal R}$, the equation

$$ax=y$$

has $n(a)=:\prod_pp^{{\rm ord}_p(a_p)}$ solutions in $x\in A_f/{\mathcal R}$. Denote these solutions by $[x:ax=y]$.

Let ${\mathbb C}[A_f/{\mathcal R}]=:{\rm span}\{\delta_x:x\in A_f/{\mathcal R}\}$ be the group algebra of $A_f/{\mathcal R}$ over ${\mathbb C}$, so that $\delta_x\delta_{x'}= \delta_{x+x'}$ for $x,x'\in A_f/{\mathcal R}$. We have,

Lemma 2. The formula

$$\alpha_a(\delta_y)=\frac1{n(a)}\sum_{[x:ax=y]}\delta_x$$

for $a\in I$ defines an action of $I$ by endomorphisms of $C^*(A_f/{\mathcal R})$.

We now appeal to the notion of semigroup crossed product developed by Laca and Raeburn, applying it to our situation. A covariant representation of $(C^*(A_f/{\mathcal R}),I,\alpha)$ is a pair $(\pi,V)$ where

$$\pi:C^*(A_f/{\mathcal R})\rightarrow B({\mathcal H})$$

is a unital representation and

$$V:I\rightarrow B({\mathcal H})$$

is an isometric representation in the bounded operators in a Hilbert space $\mathcal H$. The pair $(\pi,V)$ is required to satisfy,

$$\pi(\alpha_a(f))=V_a\pi(f)V_a^*,\qquad a\in I,\quad f\in C^*(A_f/{\mathcal R}).$$

Such a representation is given by $(\lambda, L)$ on $l^2(A_f/{\mathcal R})$ with orthonormal basis $\{e_x:x\in A_f/{\mathcal R}\}$ where $\lambda$ is the left regular representation of $C^*(A_f/{\mathcal R})$ on $l^2(A_f/{\mathcal R})$ and

$$L_ae_y=\frac1{\sqrt {n(a)}}\sum_{[x:ax=y]}e_x.$$

The universal covariant representation, through which all other covariant representations factor, is called the (semigroup) crossed product $C^*(A_f/{\mathcal R})\rtimes_{\alpha}I$. This algebra is the universal $C^*$-algebra generated by the symbols $\{e(x):x\in A_f/{\mathcal R}\}$ and $\{\mu_a:a\in I\}$ subject to the relations

$$\mu_a^*\mu_a=1,\quad\mu_a\mu_b=\mu_{ab},\qquad a,b\in I,$$ (3)

$$e(0)=1,\quad e(x)^*=e(-x),\quad e(x)e(y)=e(x+y),\qquad x,y\in A_f/{\mathcal R},$$ (4)

$$\frac1{n(a)}\sum_{[x:ax=y]}e(x)=\mu_ae(y)\mu_a^*,\qquad a\in I,y\in A_f/{\mathcal R}.$$ (5)

When $u\in W$ then $\mu_u$ is unitary, so that $\mu^*_u\mu_u=\mu_u\mu^*_u=1$ and we have for all $x\in A_f/{\mathcal R}$,

$$\mu_ue(x)\mu_u^*=e(u^{-1}x),\qquad\mu_u^*e(x)\mu_u=e(ux).$$ (6)

Therefore we have a natural action of $W$ as inner automorphisms of $C^*(A_f/{\mathcal R})\rtimes_\alpha I$.

To recover the $C^*$-algebra of Bost-Connes we must split the above short exact sequence. Let $m{\mathbb Z}$, $m\in{\mathbb Z}$, be an ideal in ${\mathbb Z}$. This generator $m$ is determined up to sign. Consider the image of $\vert m\vert$ in $I$ under the diagonal embedding $q\mapsto(q)_p$ of ${\mathbb Q}^*$ into $I$, where the $p$-th component of $(q)_p$ is the image of $q$ in ${\mathbb Q}^*_p$ under the natural embedding of ${\mathbb Q}^*$ into ${\mathbb Q}^*_p$. The map

$$+:m{\mathbb Z}\mapsto (\vert m\vert)_p$$ (7)

defines a splitting of the short exact sequence. Let $I_+$ denote the image and define $B$ to be the semigroup crossed product $C^*(A_f/{\mathcal R})\rtimes_{\alpha}I_+$ with the restricted action $\alpha$ from $I$ to $I_+$. By transport of structure, this algebra is easily seen to be isomorphic to a semigroup crossed product of $C^*({\mathbb Q}/{\mathbb Z})$ by ${\mathbb N}_+$, where ${\mathbb N}_+$ denotes the positive natural numbers. This is the algebra of Bost-Connes. The replacement of $I$ by $I_+$ now means that the group $W$ acts by outer automorphisms. For $x\in B$, one has that $\mu_u^*x\mu_u$ is still in $B$ (computing in the larger algebra $C^*(A_f/{\mathcal R})\rtimes_{\alpha}I$), but now this defines an outer action of $W$. This coincides with the definition of $W$ as the symmetry group as in the paper of Bost-Connes.

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