Journal of Operator Theory

Volume 85, Issue 2, Spring 2021 pp. 391-402.

A note on crossed products of rotation algebras

Authors: Christian Bonicke (1), Sayan Chakraborty (2), Zhuofeng He (3), Hung-Chang Liao
Author institution: (1) School of Mathematics and Statistics, University of Glasgow, University Gardens, Glasgow, G12 8QQ, U.K.
(2) Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal Bypass Road, Bhopal, 462066, India
(3) Research Center for Operator Algebras, School of Mathematical Sciences, and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, P.R. China
(4) Department of Mathematics and Statistics, University of Ottawa, 150 Louis-Pasteur Private, Ottawa, ON K1N 9A7, Canada

Summary: We compute the $K$-theory of crossed products of rotation algebras $\mathcal{A}_\theta$, for any real angle $\theta$, by matrices in $\mathrm{SL}(2,\mathbb{Z})$ with infinite order. Using techniques of continuous fields, we show that the canonical inclusion of $\mathcal{A}_\theta$ into the crossed products is injective at the level of $K_0$-groups. We then give an explicit set of generators for the $K_0$-groups and compute the tracial ranges concretely.

DOI: http://dx.doi.org/10.7900/jot.2019sep08.2283
Keywords: rotation algebras, crossed products, $K$-theory

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