
\documentclass{article}


\begin{document}

\begin{rawhtml}
<h1 align=center> The C*-algebra of Bost--Connes
</h1>
\end{rawhtml}


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
$|u_p|_p=1$. 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,
\begin{equation}
1\rightarrow W\rightarrow I\rightarrow{\bold I}\rightarrow 1.
\end{equation}
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 $|u_p|_p=p^{-{\rm ord}_p(u_p)}$. By the
Strong Approximation Theorem we have
\begin{equation}
{\mathbb Q}/{\mathbb Z}\simeq A_f/{\mathcal R}\simeq\oplus_p{\mathbb
Q}_p/{\mathbb Z}_p
\end{equation}
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

\noindent{\bf Lemma 1.} {\it 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,

\noindent{\bf Lemma 2.} {\it 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
\begin{equation}
\mu_a^*\mu_a=1,\quad\mu_a\mu_b=\mu_{ab},\qquad a,b\in I,
\end{equation}
\begin{equation}
e(0)=1,\quad e(x)^*=e(-x),\quad e(x)e(y)=e(x+y),\qquad x,y\in
A_f/{\mathcal R},
\end{equation}
\begin{equation}
\frac1{n(a)}\sum_{[x:ax=y]}e(x)=\mu_ae(y)\mu_a^*,\qquad a\in
I,y\in A_f/{\mathcal R}.
\end{equation}
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 $x\in A_f/{\mathcal R}$,
\begin{equation}
\mu_ue(x)\mu_u^*=e(u^{-1}x),\qquad\mu_u^*e(x)\mu_u=e(ux).
\end{equation}
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 $|m|$ 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
\begin{equation}
+:m{\mathbb Z}\mapsto (|m|)_p
\end{equation}
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.







\begin{rawhtml}
<p> <br>
<hr>
<p> <br>
Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
\end{rawhtml}

\end{document}

