# The C*-algebra of Bost--Connes

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)

The map in this short exact sequence is given as follows. To associate the ideal where is determined by the formula . By the Strong Approximation Theorem we have
 (2)

and we have therefore a natural action of on by multiplication in and transport of structure. We have the following straightforward Lemmata

Lemma 1. For and , the equation

has solutions in . Denote these solutions by .

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

Lemma 2. The formula

for defines an action of by endomorphisms of .

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)

When then is unitary, so that and we have for all ,
 (6)

Therefore we have a natural action of as inner automorphisms of .

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)

defines a splitting of the short exact sequence. Let denote the image and define to be the semigroup crossed product with the restricted action from to . By transport of structure, this algebra is easily seen to be isomorphic to a semigroup crossed product of by , where denotes the positive natural numbers. This is the algebra of Bost-Connes. The replacement of by now means that the group acts by outer automorphisms. For , one has that is still in (computing in the larger algebra ), but now this defines an outer action of . This coincides with the definition of as the symmetry group as in the paper of Bost-Connes.

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