Some remarks on $\mathrm{K}_0$ of noncommutative tori

  • Sayan Chakraborty

Abstract

Using Rieffel's construction of projective modules over higher dimensional noncommutative tori, we construct projective modules over some continuous field of C*-algebras whose fibres are noncommutative tori. Using a result of Echterhoff et al., our construction gives generators of $\mathrm {K}_0$ for all noncommutative tori.

References

Echterhoff, S., Lück, W., Phillips, N. C., and Walters, S., The structure of crossed products of irrational rotation algebras by finite subgroups of $\mathrm SL_2(\mathbb Z)$, J. Reine Angew. Math. 639 (2010), 173–221. https://doi.org/10.1515/CRELLE.2010.015

Elliott, G. A., On the $K$-theory of the $C^\ast $-algebra generated by a projective representation of a torsion-free discrete abelian group, in “Operator algebras and group representations, Vol. I (Neptun, 1980)”, Monogr. Stud. Math., vol. 17, Pitman, Boston, MA, 1984, pp. 157–184.

Elliott, G. A. and Li, H., Strong Morita equivalence of higher-dimensional noncommutative tori. II, Math. Ann. 341 (2008), no. 4, 825–844. https://doi.org/10.1007/s00208-008-0213-8

Li, H., Strong Morita equivalence of higher-dimensional noncommutative tori, J. Reine Angew. Math. 576 (2004), 167–180. https://doi.org/10.1515/crll.2004.087

Prodan, E. and Schulz-Baldes, H., Bulk and boundary invariants for complex topological insulators: from $K$-theory to physics, Mathematical Physics Studies, Springer, 2016. https://doi.org/10.1007/978-3-319-29351-6

Renault, J., A groupoid approach to $C^\ast $-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980.

Rieffel, M. A., Projective modules over higher-dimensional noncommutative tori, Canad. J. Math. 40 (1988), no. 2, 257–338. https://doi.org/10.4153/CJM-1988-012-9
Published
2020-05-06
How to Cite
Chakraborty, S. (2020). Some remarks on $\mathrm{K}_0$ of noncommutative tori. MATHEMATICA SCANDINAVICA, 126(2), 387-400. https://doi.org/10.7146/math.scand.a-119699
Section
Articles