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

TL;DR

This paper characterizes the C*-algebra associated with an integral domain as a partial crossed product, revealing its structure and properties like simplicity through topological dynamics.

Contribution

It provides a new description of the C*-algebra A[R] as a partial group algebra and reconstructs known results using partial crossed product theory.

Findings

A[R] is a partial group algebra of K ⋉ K^x with relations.

The spectrum of relations is homeomorphic to the profinite completion of R.

A[R] is simple due to topologically free and minimal action.

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 $K \rtimes K^x$ 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.