We give a different construction of the -algebra of
Bost-Connes to that found in their original paper. It is directly
inspired by work of Arledge-Laca-Raeburn. Let denote the
ring of finite adeles of , that is the restricted
product of with respect to as
ranges over the finite primes. Recall that this restricted product
consists of the infinite vectors , indexed by the primes
, such that
with
for almost all primes . The group of (finite) ideles
consists of the invertible elements of the adeles. Let
be those elements of
with
. Notice that an idele has
with
for almost all primes .
Let

Further, let denoted the semigroup of integral ideals of , which are of the form where . Notice that as above is also a semigroup. We have a natural short exact sequence,

(1) |

(2) |

**Lemma 1.** *For
and
, the
equation
*

Let be the group algebra of over , so that for . We have,

**Lemma 2.** *The formula
*

We now appeal to the notion of semigroup crossed product developed
by Laca and Raeburn, applying it to our situation. A covariant
representation of
is a pair
where

is a unital representation and

is an isometric representation in the bounded operators in a Hilbert space . The pair is required to satisfy,

Such a representation is given by on with orthonormal basis where is the left regular representation of on and

The universal covariant representation, through which all other covariant representations factor, is called the (semigroup) crossed product . This algebra is the universal -algebra generated by the symbols and subject to the relations

(3) |

(4) |

(5) |

(6) |

To recover the -algebra of Bost-Connes we must split the
above short exact sequence. Let ,
,
be an ideal in . This generator is determined up
to sign. Consider the image of in under the diagonal
embedding of into , where the
-th component of is the image of in
under the natural embedding of into
. The map

(7) |

Back to the
main index
for The Riemann Hypothesis.