Projections in Rotation Algebras and Theta Functions

For each $\alpha \in (0,1)$, $A_{\alpha }$ denotes the universal $C^{\ast }$-algebra generated by two unitaries $u$ and $v$, which satisfy the commutation relation $uv=\exp (2\pi i\alpha )vu$. We consider the order four automorphism $\sigma$ of $A_{\alpha }$ defined by $\sigma (u)=v$, $\sigma (v)=u^{-1}$ and describe a method for constructing projections in the fixed point algebra $A_{\alpha }^{\sigma }$, using Rieffel’s imprimitivity bimodules and Jacobi’s theta functions. In the case $\alpha =q^{-1}$, $q\in \mathbf{Z}$, $q\ge 2$, we give explicit formulae for such projections and find a lower bound for the norm of the Harper operator $u+u^{\ast }+v+v^{\ast }$.