# Journal of Operator Theory

Volume 41, Issue 2, Spring 1999 pp. 291-319.

K-theory for $C^*$-algebras associated to lattices in Heisenberg Lie groups**Authors**: Soo Teck Lee (1), and Judith A. Packer (2)

**Author institution:**(1) Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Republic of Singapore

(2) Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Republic of Singapore

**Summary:**We present methods for computing the $\K$-groups of a variety of $C^\ast$-algebras associated to lattices in Heisenberg Lie groups, including twisted group $C^\ast$-algebras and Azumaya algebras over the corresponding nilmanifolds. A precise formula for the rank of the above $\K$-groups is given, and it is shown that any twisted group $C^\ast$-algebra over such a lattice $\Gamma$ is $\KK$-equivalent to an ordinary group $C^\ast$-algebra corresponding to a possibly different lattice $\Gamma_{\scriptscriptstyle 0}$. We also give applications of our methods to the calculation of $\K$-groups for certain twisted transformation group $C^\ast$-algebras and certain continuous trace algebras whose spectra are tori.

**Keywords:**Discrete Heisenberg groups, $C^*\!$-algebras, Brauer group, {\rm K}$\!$-theory

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