Partial crossed product description of the C*-algebras associated with integral domains

Recently, Cuntz and Li introduced the C^*-algebra A[R] associated to an integral domain R with finite quotients. In this paper, we show that A[R] is a partial group algebra of the group $K \rtimes K^x$ with suitable relations, where K is the field of fractions of R. We identify the spectrum of this relations and we show that it is homeomorphic to the profinite completion of R. By using partial crossed product theory, we reconstruct some results proved by Cuntz and Li. Among them, we prove that $\ar$ is simple by showing that the action is topologically free and minimal.


Introduction
Fifteen years ago, motivated by the work of Julia [14], Bost and Connes constructed a C * -dynamical system having the Riemann ζ-function as partition function [2]. The C *algebra of the Bost-Connes system, denoted by C Q , is a Hecke C * -algebra obtained from the inclusion of the integers into the rational numbers. In [19], Laca and Raeburn showed that C Q can be realized as a semigroup crossed product and, in [20], they characterized the primitive ideal space of C Q .
In [1], [4] and [15], by observing that the construction of C Q is based on the inclusion of the integers into the rational numbers, Arledge, Cohen, Laca and Raeburn generalized the construction of Bost and Connes. They replaced the field Q by an algebraic number field K and Z by the ring of integers of K. Many of the results obtained for C Q were generalized to arbitrary algebraic number fields (at least when the ideal class group of the field is h = 1) [16], [17].
Recently, a new construction appeared. In [5], Cuntz defined two new C * -algebras: Q N and Q Z . Both algebras are simple and purely infinite and Q N can be seen as a C * -subalgebra of Q Z . These algebras encode the additive and multiplicative structure of the semiring N and of the ring Z. Cuntz showed that the algebra Q N is, essentially, the algebra generated by C Q and one unitary operator. In [25], Yamashita realized Q N as the C * -algebra of a topological higher-rank graph.
The next step was given by Cuntz and Li. In [6], they generalized the construction of Q Z by replacing Z by a unital commutative ring R (which is an integral domain with finite quotients by principal ideals). This algebra was called A[R]. Cuntz and Li showed that A[R] is simple and purely infinite (when R is not a field) and they related a C * -subalgebra of its with the generalized Bost-Connes systems (when R is the ring of integers in an algebraic number field having h = 1 and, at most, one real place). In [23], Li extended the construction of A[R] to an arbitrary unital ring.
The aim of this text is to show that the algebra A[R] can be seen as a partial crossed product (when R is an integral domain with finite quotients). We show that A[R] is isomorphic to a partial group algebra of the group K ⋊ K × with suitable relations, where K is the field of fractions of R. By using the relationship between partial group algebras and partial crossed products, we see that A[R] is a partial crossed product by the group K ⋊ K × . We characterize the spectrum of the commutative algebra arising in the crossed product and show that this spectrum is homeomorphic toR (the profinite completion of R). Furthermore, we show that the partial action is topologically free and minimal. By using that the group K ⋊ K × is amenable, we conclude that A[R] is simple.
Recently, some similar results appeared. In [21] and [3], Brownlowe, an Huef, Laca and Raeburn showed that Q N is a partial crossed product by using a boundary quotient of the Toeplitz (or Wiener-Hopf) algebra of the quasi-lattice ordered group (Q ⋊ Q × + , N ⋊ N × ) (see [24] and [18] for Toeplitz algebras of quasi-lattice ordered groups). We observe that our techniques are different from theirs. We don't use Nica's construction [24] (indeed, our group K ⋊ K × is not a quasi-lattice, in general). From our results, in the case R = Z, we see that Q Z is a partial crossed product by the group Q ⋊ Q × . From this, it's imediate that Q N is a partial crossed product by Q ⋊ Q × + (as in [3]). Before we go to the main result we give, in the section 2, a breifly review about the algebra A[R] and the theories of partial crossed products and partial group algebras. In the section 3, we state our main theorem: the algebra A[R] is isomorphic to a partial group algebra. In the section 4, we study A[R] by using the techniques of partial crossed products. We recover the faithful conditional expectation constructed by Cuntz and Li in [6, Proposition 1] in a very natural way. Futhermore, we use the concepts of topological freeness and minimality of a partial action to show that A[R] is simple.

Preliminaries
2.1. The C * -algebra A[R] of an Integral Domain. Throughout this text, R will be an integral domain (unital commutative ring without zero divisors) with the property that the quotient R/(m) is finite, for all m = 0 in R. We denote by R × the set R\{0} and by R * the set of units in R.
, is the universal C*-algebra generated by isometries {s m | m ∈ R × } and unitaries {u n | n ∈ R} subject to the relations for all m, m ′ ∈ R × and n, n ′ ∈ R.
We denote by e m the range projection of s m , namely e m = s m s * m . It's easily seen that, under (CL2) and (CL3), u l e m u −l = u l ′ e m u −l ′ if l + (m) = l ′ + (m). From this, we see that the sum in (CL4) is independent of the choice of l.
Let {ξ r | r ∈ R} be the canonical basis of the Hilbert space ℓ 2 (R) and consider the operators S m and U n on ℓ 2 (R) given by S m (ξ r ) = ξ mr and U n (ξ r ) = ξ n+r .
Definition 2.2. [6, Section 2] The reduced regular C * -algebra of R, denoted by A r [R], is the C * -subalgebra of B(ℓ 2 (R)) generated by the operators {S m | m ∈ R × } and {U n | n ∈ R}.
One can checks that S m is an isometry, U n is a unitary and satisfy (CL1)-(CL4). Hence, there exists a surjective * -homomorphism In [6], Cuntz and Li showed that, when R is not a field, A[R] is simple; therefore the above * -homomorphism is a * -isomorphism. In the section 4, we will show that A[R] is simple (when R is not a field) by using the partial crossed product description of A[R].

Partial Crossed Products.
Here, we review some basic facts about partial actions and partial crossed products.
Definition 2.4. [9, Definition 1.1] A partial action α of a (discrete) group G on a C * -algebra A is a collection (D g ) g∈G of ideals of A and * -isomorphisms α g : where e represents the identity element of G; . In the above definition, if we replace the C * -algebra A by a locally compact space X, the ideals D g by open sets X g and the * -isomorphisms α g by homeomorphisms θ g : X g −1 −→ X g , we obtain a partial action θ of the group G on the space X. A partial action θ on a space X induces naturally a partial action α on the C * -algebra C 0 (X). The ideals D g are C 0 (X g ) and α g (f ) = f • θ g −1 .
We say that a partial action θ on a space X is topologically free if, for all g ∈ G\{e}, the set F g = {x ∈ X g −1 | θ g (x) = x} has empty interior. A subset V of X is invariant under the partial action θ if θ g (V ∩ X g −1 ) ⊆ V , for every g ∈ G. The partial action θ is minimal if there are no invariant open subsets of X other than ∅ and X. It's easy to see that θ is minimal if, and only if, every x ∈ X has dense orbit, namely From a partial action α, we can construct two partial crossed products: A ⋊ α G (full) and A ⋊ α,r G (reduced). We can define both as follows: let L be the normed * -algebra of the finite formal sums g∈G a g δ g , where a g ∈ D g . The operations and the norm in L are given by (a g δ g )(a h δ h ) = α g (α g −1 (a g )a h )δ gh , (a g δ g ) * = α g −1 (a * g )δ g −1 and || g∈G a g δ g || = g∈G ||a g ||. If we denote by B g the vector subspace D g δ g of L, then the family (B g ) g∈G generates a Fell bundle. The full and the reduced crossed products are, respectively, the full and the reduced cross sectional algebra of (B g ) g∈G . It's well known that A ⋊ α G is universal with respect to a covariant pair (ϕ, π), where ϕ : There exists a faithful conditional expectation E : A ⋊ α,r G −→ A given by E(aδ g ) = a if g = e, and E(aδ g ) = 0 if g = e. When the Fell bundle (B g ) g∈G is amenable (G amenable implies its), the full and reduced constructions are isomorphic and, in this case, there exists a faithful conditional expectation of There is a close relation between topological freeness and minimality of the partial action and ideals of the reduced crossed product. If θ is a topologically free partial action on a space X then θ is minimal if, and only if, C 0 (X) ⋊ α,r G is simple, where α is the action induced by θ. Under the amenability hypothesis, this result is valid for the full crossed product too.
2.3. Partial Group Algebras. Let G be a discrete group, let G be the set G without the group operations and denote the elements in G by [g] (namely, The partial group algebra of G, denoted by C * p (G), is defined to be the universal C * -algebra generated by the set G with the relations The algebra C * p (G) is universal with respect to a partial representation. Observe that the relations in R p correspond to the partial representation axioms (PR1), (PR2) and (PR3). Sometimes, we will refer to a relation in R p by indicating the corresponding axiom.
Consider the natural bijection between P(G) and {0, 1} G , where P(G) is the power set of G. With the product topology, {0, 1} G is a compact Hausdorff space. Give to P(G) the topology of {0, 1} G . Denote by X G the subset of P(G) of the subsets ξ of G such that e ∈ ξ. Clearly, with the induced topology of P(G), X G is a compact space.
It's easy to see that θ g : X g −1 −→ X g given by θ g (ξ) = gξ is a homeomorphism. The collection of open sets (X g ) g∈G of X G with the homeomorphisms θ g define a partial action θ of G on X G . The partial crossed product The partial group algebra of G with relations R, denoted by C * p (G, R), is defined to be the universal C * -algebra generated by the set G with the relations R p ∪ R. Given a partial representation π of G, we can extend π naturally to sums of produtcs of elements in G. If this extension satisfies the relations R, we say that π is a partial representation that satisfies R. The algebra C * p (G, R) is universal with respect to a partial representation that satisfies the relations R.
Denote by 1 g the function in C(X G ) given by 1 g (ξ) = 1 if g ∈ ξ and 1 g (ξ) = 0 otherwise. By an abuse of notation, we also denote by R the subset of C(X G ) given by the functions i λ i j 1 g ij , where i λ i j e g ij = 0 is a relation in (the original) R. The spectrum of the relations R is defined to be the compact Hausdorff space Let Ω g = {ξ ∈ Ω R | g ∈ ξ}. By restricting the above θ g to Ω g −1 , we obtain a partial action (again denoted by θ) of G on Ω R (the open sets are the Ω g 's and the homeomorphisms are the restrictions of the θ g 's). The main result concerning C * p (G, R) says that this algebra is isomorphic to the partial crossed product C(Ω R ) ⋊ α G (again, α is the partial action induced by θ).
The above results are proved in [12] and [13].

Partial Group Algebra Description of A[R]
Let R be an integral domain satisfying the conditions stated in the previous section. Denote by K the field of fractions of R and consider the semidirect product K ⋊ K × . The elements of K ⋊ K × will be denoted by a pair (u, w), where u ∈ K and w ∈ K × . Recall that (u, w)(u ′ , w ′ ) = (u + u ′ w, ww ′ ) and (u, w) −1 = (−u/w, 1/w). We denote by [u, w] an element of set K ⋊K × without the group operations (as the set G associated to G in the previous section). 1    Consider the partial group algebra C * p (K ⋊ K × , R). We will show that this algebra is isomorphic to A[R].  Proof. Let n m ′ , m m ′ = q p ′ , p p ′ , ie, pm ′ = p ′ m and m ′ q = p ′ n. Hence, Proposition 3.4. The map π defined above is a partial representation of K ⋊ K × that satisfies R.
Proof. First, we will show that π is a partial representation. Since π((0, 1)) = s * 1 u 0 s 1 = 1, we have (PR1). Observe that . This shows that π is a partial representation. It remains to show that the extention of π satisfies the relations in R. By remark 3.1, it suffices to show that the relations in

Partial Crossed Product Description of A[R]
Before characterizing A[R] as a partial crossed product, note that the group K ⋊ K × is solvable and, hence, amenable. Therefore, there exists a faithful conditional expectation (imported from the partial crossed product realization) E : In On the other hand We already know that A[R] is a partial crossed product. Indeed, every partial group algebra is a partial crossed product (see section 2.3). From now on, our goal is to study A[R] by this way.
There exists a natural partial order on R × given by the divisibility: we say that m ≤ m ′ if there exists r ∈ R such that m ′ = mr. Whenever m ≤ m ′ , we can consider the canonical projection p m,m ′ : R/(m ′ ) −→ R/(m). Since (R × , ≤) is a directed set, we can consider the inverse limitR which is the profinite completion of R. In this text, we shall use the following concrete description ofR: Give to R/(m) the discrete topology, to m∈R × R/(m) the product topology and toR the induced topology of m∈R × R/(m). With the operations defined componentwise, R is a compact topological ring. There exists a canonical inclusion of R intoR given by r −→ (r + (m)) m (to see injectivity, take r = 0, m non-invertible and note that r / ∈ (rm)). The above partial order can be extended to K × . For w, w ′ ∈ K × , we say that w ≤ w ′ if there exists r ∈ R such that w ′ = wr. Denote by (w) the fractional ideal generated by w, namely (w) = wR ⊆ K. As before, if w ≤ w ′ , we can consider the canonical projection 3 p w,w ′ : (R + (w ′ ))/(w ′ ) −→ (R + (w))/(w). As before, we consider the inverse limit It is a compact topological ring too. In fact,R K is naturally isomorphic toR as topological ring. In this text, we useR K instead ofR to simplify our proofs.
It's easy to see that, when R is a field, thenR ∼ =R K ∼ = {0}.
Let Ω be the spectrum of the relations R (see section 2.3). We will show that Ω is homeomorphic toR K (hence, homeomorphic toR). Define Note that the definition is independent of the choice of u w in u w + (w).
Following the section 2.3, there exists a partial action of K ⋊ K × on Ω. By the above proposition, we can define this partial action onR K . LetR g = ρ −1 (Ω g ), where Ω g = {ξ ∈ Ω | g ∈ ξ}, and θ g be the homeomorphism betweenR g −1 andR g . It's easy to see thatR ie, θ (u,w) acts onR (u,w) −1 by the affine transformation corresponding to (u, w). The next proposition, whose proof is trivial, will be useful later.
Proposition 4.4. We have that (i)R (u,w) = ∅ ⇐⇒ u / ∈ R + (w); (ii)R (u,w) =R K ⇐⇒ R ⊆ u + (w). Now, we describe the topology onR K . SinceR K is a singleton set when R is a field, we shall assume that R is not a field in this paragraph. For w ∈ K × and C w ⊆ (R + (w))/(w), we define the open set ). Futhermore, if C w = ∅, r is a non-invertible element in R and V Cw w = V Cwr wr , then C wr has, at least, two elements. Indeed, let u + (w) ∈ C w and r 1 , r 2 ∈ R such that r 1 + (r) = r 2 + (r). It's easy to see that u + wr 1 + (wr) and u + wr 2 + (wr) are in C wr and that u + wr 1 + (wr) = u + wr 2 + (wr). This says that, if V Cw w is non-empty, we can suppose that C w has more than one element.
Proposition 4.5. The partial action θ onR K is topologically free if, and only if, R is not a field.
Proposition 4.6. The partial action θ is minimal.
Proof. If R is a field, then the result is trivial. Now, suppose that R is not a field. We will prove that every x ∈R K has dense orbit (see section 2.2) by showing that if V is a non-empty open set, then there exists g ∈ K ⋊ K × such that x ∈R g −1 and θ g (x) ∈ V . Let x = (u w + (w)) w ∈R K and V = V C w ′ w ′ non-empty. Take u ′ + (w ′ ) ∈ C w ′ and observe that we can suppose, without loss of generality, u ′ ∈ R and u w ′ ∈ R. Let g = (u ′ − u w ′ , 1). By the proposition 4.4,R g −1 =R K and, hence, x ∈R g −1 . To finish, note that θ g (x) = θ (u ′ −u w ′ ,1) ((u w + (w)) w ) = (u ′ − u w ′ + u w + (w)) w ∈ V .
Following, we summarize the results of this section.
Theorem 4.7. The algebra A[R] is * -isomorphic to the partial crossed product C(R K )⋊ α K ⋊ K × , where α is the partial action induced by θ. The * -isomorphism is given by u n −→ 1δ (n,1) and s m −→ 1 (0,m) δ (0,m) , where 1 (0,m) is the characteristic function ofR g . Proof. By the propositions 4.5 and 4.6, the reduced crossed product C(R K ) ⋊ α,r K ⋊K × is simple. Since K ⋊ K × is amenable, then C(R K ) ⋊ α K ⋊ K × ∼ = C(R K ) ⋊ α,r K ⋊ K × and, therefore, C(R K ) ⋊ α K ⋊ K × is simple. The result follows from the previous theorem. When R = Z, we can restrict our partial action to the subgroup Q ⋊ Q * + of Q ⋊ Q * and the corresponding partial crossed product is the algebra Q N introduced by Cuntz in [5] and realized as a partial crossed product in [3] by Brownlowe, an Huef, Laca and Raeburn.