Partial Crossed Product Description of the Cuntz-Li Algebras
Cuntz-Li algebras | ring C*-algebras | Bost-Connes algebra | partial group algebra | partial crossed product.
In this text, we study three algebras: Cuntz-Li, ring and Bost-Connes algebras. The first one, presented by Cuntz and Li in 2008 and denoted by A[R], are C*-algebras associated to an integral domain R with finite quotients. We show that A[R] is a partial group algebra with suitable relations and we identify the spectrum of these relations as 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. In 2009, Li generalized the Cuntz-Li algebras for more general rings and called it ring C*-algebras. Here, we propose a new extension for the Cuntz-Li algebras. Unlike ring C*-algebras, our construction takes into account the zero-divisors of the ring in definition of the multiplication operators. In 1995, Bost and Connes constructed a C*-dynamical system having the Riemann zeta function as partition function. We conclude this work proving that the C*-algebra underlying the Bost-Connes system has a partial crossed product structure.