Journal of Operator Theory

Volume 39, Issue 2, Spring 1998 pp. 395-400.

The automorphism groups of rational rotation algebras

Authors: P.J. Stacey
Author institution: School of Mathematics, La Trobe University, Bundoora, Victoria 3083, Australia

Summary: $\def\Homeo{{\rm Homeo}\,} \def\Aut{{\rm Aut}\,} \def\Inn{{\rm Inn}\,}$ Let $A_\theta$ be the universal $C^*$-algebra generated by two unitaries $U, \ V$ with $VU = \rho UV$, where $\rho = {\rm e}^{2\pi{\rm i}\theta}$ and $\theta $ is rational. Let Aut$A_\theta$ be the group of $*$-automorphisms of $A_\theta$. It is shown that if $\theta \not= \frac 12$ then the image of the natural map from Aut$A_\theta$ to Homeo${\mathbb T}^2$ is the subgroup Homeo$_+{\mathbb T}^2$ of orient ation preserving homeomorphisms of the torus ${\mathbb T}^2$. Hence there exist exact sequences $$0 \to \Inn A_\theta \to \Aut A_\theta \to \Homeo {\mathbb T}^2 \to 0 $$ when $\theta = \frac 12$ and $$ 0 \to \Inn A_\theta \to \Aut A_\theta \to \Homeo_+ {\mathbb T}^2 \to 0 $$ when $\theta \not= \frac 12$, where Inn$A_\theta$ is the group of inner automorphisms.

Keywords: $C^*$-algebra, rational rotation algebra, automorphisms

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