Projections in Rotation Algebras and Theta Functions

Details Projections in Rotation Algebras and Theta Functions Projections in Rotation Algebras and Theta Functions

1998-03-27

arxiv.org/abs/math/9803134v1

Mathematics

Operator Algebras

27 pages

Scientific paper

10.1007/s002200050585

For each $\alpha \in (0,1)$, $A_\alpha$ denotes the universal $C^*$-algebra generated by two unitaries $u$ and $v$, which fulfill 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 $\alpha =q^{-1}$, $q\in {\bf Z}$, $q\geq 2$, we give explicit formulae for such projections and find some lower bounds for $|u+u^* +v+v^*|$.

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