\(C^*\)-algebras associated with the \(ax+b\)-semigroup over \(\mathbb N\). (English) Zbl 1162.46036
Cortiñas, Guillermo (ed.) et al., \(K\)-theory and noncommutative geometry. Proceedings of the ICM 2006 satellite conference, Valladolid, Spain, August 31–September 6, 2006. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-060-9/hbk). EMS Series of Congress Reports, 201-215 (2008).
Summary: We present a \(C^*\)-algebra (denoted by \({\mathcal Q}_{\mathbb N})\) which is associated to the \(ax + b\)-semigroup over \(\mathbb N\). It is, in fact, a natural quotient of the \(C^*\)-algebra associated with this semigroup by some additional relations (which make it simple and purely infinite). These relations are satisfied in representations related to number theory. The study of this algebra is motivated by the construction in [J.-B. Bost and A. Connes, Sel. Math., New Ser. 1, No. 3, 411–457 (1995; Zbl 0842.46040)]. Our \(C^*\)-algebra contains the algebra considered by Bost and Connes, but in addition a generator corresponding to translation by the additive group \(\mathbb Z\).
As a \(C^*\)-algebra, \({\mathcal Q}_{\mathbb N}\) has an interesting structure. It is a crossed product of the Bunce-Deddens algebra associated to \(\mathbb Q\) by the action of the multiplicative semigroup \(\mathbb N^\times \). It has a unique canonical KMS-state. We also determine its \(K\)-theory, whose generators turn out to be determined by prime numbers.
On the other hand, \({\mathcal Q}_{\mathbb N}\) can also be obtained as a crossed product of the commutative algebra of continuous functions on the completion \(\widehat{\mathbb Z}\) by the natural action of the \(ax+b\)-semigroup over \(\mathbb N\).
In the last section, we consider the analogous construction of a \(C^*\)-algebra replacing the multiplicative semigroup \(\mathbb N^\times\) by \(\mathbb Z^\times\).
For the entire collection see [Zbl 1147.19001].
As a \(C^*\)-algebra, \({\mathcal Q}_{\mathbb N}\) has an interesting structure. It is a crossed product of the Bunce-Deddens algebra associated to \(\mathbb Q\) by the action of the multiplicative semigroup \(\mathbb N^\times \). It has a unique canonical KMS-state. We also determine its \(K\)-theory, whose generators turn out to be determined by prime numbers.
On the other hand, \({\mathcal Q}_{\mathbb N}\) can also be obtained as a crossed product of the commutative algebra of continuous functions on the completion \(\widehat{\mathbb Z}\) by the natural action of the \(ax+b\)-semigroup over \(\mathbb N\).
In the last section, we consider the analogous construction of a \(C^*\)-algebra replacing the multiplicative semigroup \(\mathbb N^\times\) by \(\mathbb Z^\times\).
For the entire collection see [Zbl 1147.19001].
