  For each $\alpha \in (0,1)$, $A_\alpha$ denotes the universal $C^*$-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 $\alpha =q^{-1}$, $q\in {\mathbf Z}$, $q\geq 2$, we give explicit formulae
for such projections and find a lower bound for the norm of the Harper operator
$u+u^* +v+v^*$.
