The bundle structure of noncommutative tori over UHF-algebras. (English) Zbl 1049.46053

Park, Chun-Gil

Bull. Belg. Math. Soc. - Simon Stevin 10, No. 3, 321-328 (2003).

The author discusses the noncommutative torus of rank \(n\) as the \(C^*\)-algebra of sections of a locally trivial continuous \(C^*\)-algebra bundle for some totally skew multiplier. He shows that the typical fiber is isomorphic to the tensor product of some completely irrational noncommutative torus and the algebra of complex \(k\times k\)-matrices for some \(k.\) The author proves that in the case where \(k\) is replaced by \(l^\infty\) for some \(l\), the corresponding tensor product has a trivial bundle structure if and only if the set of prime factors of \(k\) is a subset of the set of prime factors of \(l.\) This result is then applied to discuss the bundle structure of the tensor products of Cuntz algebras with noncommutative tori.

Reviewer: Jürgen Tolksdorf (Leipzig)

MSC:

46L87	Noncommutative differential geometry
46L05	General theory of \(C^*\)-algebras

Keywords:

noncommutative torus; Cuntz algebras