Projections in Rotation Algebras and Theta Functions

Abstract

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