Tutorial on rational rotation $C^*$-algebras

The rotation algebra $\mathcal A_{\theta}$ is the universal $C^*$-algebra generated by unitary operators $U, V$ satisfying the commutation relation $UV = \omega V U$ where $\omega= e^{2\pi i \theta}.$ They are rational if $\theta = p/q$ with $1 \leqslant p \leqslant q-1,$ othewise irrational. Operators in these algebras relate to the quantum Hall effect [2,26,30], kicked quantum systems [22, 34], and the spectacular solution of the Ten Martini problem [1]. Brabanter [4] and Yin [38] classified rational rotation $C^*$-algebras up to $*$-isomorphism. Stacey [31] constructed their automorphism groups. They used methods known to experts: cocycles, crossed products, Dixmier-Douady classes, ergodic actions, $\mathrm{K}$-theory, and Morita equivalence. This expository paper defines $\mathcal A_{p/q}$ as a $C^*$-algebra generated by two operators on a Hilbert space and uses linear algebra, Fourier series and the Gelfand–Naimark–Segal construction [16] to prove its universality. It then represents it as the algebra of sections of a matrix algebra bundle over a torus to compute its isomorphism class. The remarks section relates these concepts to general operator algebra theory. We write for mathematicians who are not $C^*$-algebra experts.