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

Abstract

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 ⋊ K × 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 A[R] is simple by showing that the action is topologically free and minimal.

Purely infinite partial crossed products

“…We conclude that U = X g i is (G, τ X )-paradoxical. for m, m ′ ∈ R × and n, n ′ ∈ R. Following Boava and Exel [5], let K denote the field of fractions of R, and K × the set K\{0}. Let G be the semidirect product K ⋊ K × = {(u, w) : u ∈ K, w ∈ K × } equipped with the following operations (u, w)(u ′ , w ′ ) = (u + u ′ w, ww ′ ) and (u, w) −1 = (−u/w, 1/w).…”

“…Let G be the semidirect product K ⋊ K × = {(u, w) : u ∈ K, w ∈ K × } equipped with the following operations (u, w)(u ′ , w ′ ) = (u + u ′ w, ww ′ ) and (u, w) −1 = (−u/w, 1/w). As in [5] we define a partial order on K × given by w ≤ w ′ if w ′ = wr for some r ∈ R. Let (w) denote the ideal wR ⊆ K.…”

“…Boava and Exel showed in [5] that the family of sets (V Cw w ) w∈K × is closed under complement, intersections and finite unions. It follows that U = V Cw w for some w ∈ K × and C w ⊆ (R+(w))/(w).…”

“…Partial actions of a discrete group on C * -algebras and their associated crossed products were gradually introduced in [14] and [35], and since then developed by many authors. Several important classes of C * -algebras have been realised as crossed products by partial actions, including in particular AF-algebras, Cuntz-Krieger algebras, Bunce-Deddens algebras, among others (see for example [5,12,13,15,16,19,26]). The description of C * -algebras as partial crossed products has also proved useful for the computation of their K-theory.…”

Bratteli–Vershik models for partial actions of ℤ

“…In cases such as this, we still have that the Vershik map is a homeomorphism between open subsets of the path space, and we can hence study it as a partial action of Z (the Vershik map has been studied as a partially defined map before in the literature, see [23,27]). Partial actions were originally defined in [11] and gradually gained importance as many C*-algebras and algebras were realized as partial crossed products (approximately finite, Bunce-Deddens and Cuntz-Krieger algebras, graph algebras, Leavitt path algebras, algebras associated with integral domains, and self-similar graph algebras among others, see [6,8,12,13,14,16,24]). In the topological category a partial action of Z provides the correct setting to study the dynamics of a partial homeomorphism: given a homeomorphism h : U → V between two open sets of the topological space X one considers the iterates h n , with n ∈ Z, restricted to the appropriate domains.…”