A crossed-product approach to the Cuntz-Li algebras (Q2891993)

The authors show that the Cuntz-Li algebra \(\mathfrak{A}[R]\) associated to an integral domain \(R\), originally defined via generators and relations, shown to be purely infinite and simple, and identified as a full corner in a group crossed product \(C^*\)-algebra by \textit{J. Cuntz} and \textit{X. Li}, [Quanta of maths. Conference on non commutative geometry in honor of Alain Connes, Paris, France, March 29--April 6, 2007. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute. Clay Mathematics Proceedings 11, 149--170 (2010; Zbl 1219.46059)], can be obtained from a group crossed product via an almost reverse construction. Namely, they first construct a class of group crossed products that turn out to be purely infinite and simple, and then they identify full corners for which generators and relations can be exhibited. One example of a full corner in this class is then \(\mathfrak{A}[R]\).NEWLINENEWLINEThe acting group is a semidirect product group \(G=N\rtimes H\) with an \(ax+b\)-type action on the normal subgroup \(N\) (where \(N\) acts by left-translation and \(H\) acts by conjugation), and a family \(\mathcal{U}\) of conjugates by elements in \(H\) of a normal subgroup \(M\) of \(N\) which satisfies a number of conditions. Restriction of the action to a \(C^*\)-subalgebra \(D\) of \(l^\infty(N)\) gives rise to a crossed product \(D\rtimes_r N\) that is purely infinite and simple when \(\mathcal{U}\) has finite quotients and \(H\) acts effectively on \(M\), implying the action is minimal, locally contractive and topologically free. Then purely infinite simple follows from a result of \textit{M. Laca} and \textit{J. Spielberg} [J. Reine Angew. Math. 480, 125-139 (1996Zbl 0863.46044)]. To identify generators and relations for certain corners, the authors use the general theory of dilations of isometric representations of semigroups, cf. [\textit{R. G. Douglas}, Bull. Lond. Math. Soc. 1, 157--159 (1969; Zbl 0187.06501)]. As an application of their techniques, the authors show that the Hecke \(C^*\)-algebra of the pair \((G, M)\) embeds faithfully in the Cuntz-Li type corner of \(D\rtimes_r N\).