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: 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

${\bbb T}^2$ is the subgroup Homeo

$_+{\bbb T}^2$ of orient ation preserving homeomorphisms of the torus

${\bbb T}^2$ . Hence there exist exact sequences

$0 \to \Inn A_\theta \to \Aut A_\theta \to \Homeo {\bbb T}^2 \to 0$ when

$\theta = \frac 12$ and

$0 \to \Inn A_\theta \to \Aut A_\theta \to \Homeo_+ {\bbb T}^2 \to 0$ when

$\theta \not= \frac 12$ , where Inn

$A_\theta$ is the group of inner automorphisms.

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