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Chern Characters of Fourier Modules


Published:20000601
Printed: Jun 2000
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Abstract
Let $A_\theta$ denote the rotation algebrathe universal $C^\ast$algebra
generated by unitaries $U,V$ satisfying $VU=e^{2\pi i\theta}UV$, where
$\theta$ is a fixed real number. Let $\sigma$ denote the Fourier
automorphism of $A_\theta$ defined by $U\mapsto V$, $V\mapsto U^{1}$,
and let $B_\theta = A_\theta \rtimes_\sigma \mathbb{Z}/4\mathbb{Z}$ denote
the associated $C^\ast$crossed product. It is shown that there is a
canonical inclusion $\mathbb{Z}^9 \hookrightarrow K_0(B_\theta)$ for each
$\theta$ given by nine canonical modules. The unbounded trace functionals
of $B_\theta$ (yielding the Chern characters here) are calculated to obtain
the cyclic cohomology group of order zero $\HC^0(B_\theta)$ when
$\theta$ is irrational. The Chern characters of the nine modulesand more
importantly, the Fourier moduleare computed and shown to involve techniques
from the theory of Jacobi's theta functions. Also derived are explicit
equations connecting unbounded traces across strong Morita equivalence, which
turn out to be noncommutative extensions of certain theta function equations.
These results provide the basis for showing that for a dense $G_\delta$ set
of values of $\theta$ one has $K_0(B_\theta)\cong\mathbb{Z}^9$ and is
generated by the nine classes constructed here.