Toeplitz Algebras and Extensions of\\Irrational Rotation Algebras

http://dx.doi.org/10.4153/CMB-2005-056-2
Canad. Math. Bull. 48(2005), 607-613

  Published:2005-12-01
 Printed: Dec 2005

For a given irrational number $\theta$, we define Toeplitz operators with symbols in the irrational rotation algebra ${\mathcal A}_\theta$, and we show that the $C^*$-algebra $\mathcal T({\mathcal A}_\theta)$ generated by these Toeplitz operators is an extension of ${\mathcal A}_\theta$ by the algebra of compact operators. We then use these extensions to explicitly exhibit generators of the group $KK^1({\mathcal A}_\theta,\mathbb C)$. We also prove an index theorem for $\mathcal T({\mathcal A}_\theta)$ that generalizes the standard index theorem for Toeplitz operators on the circle.