Journal of Operator Theory

Volume 57, Issue 1, Winter 2007 pp. 35-65.

Connes-Chern characters of hexic and cubic mod ules

Authors: J. Buck (1) and S. Walters (2)
Author institution: (1) Department of Mathematics, University of Oregon, Eugene OR 97403-1222, USA
(2) Department of Mathematics, Univ. of Northern British Columbia, Prince George, B.C. V2N 4Z9, Canada

Summary: Let $A_\theta$ denote the rotation $C^*$-algebra generated by unitaries $U,V$ satisfying $VU=\mathrm e^{2\pi\mathrm i\theta}UV$, where $\theta$ is a fixed rea l number. Let $\rho$ denote the {\it hexic} transform of $A_\theta$ defined by $U\mapsto V \mapsto \mathrm e^{-\pi\mathrm i\theta}U^{-1}V$ (which has order six ), let $\kappa$ denote the {\it cubic} transform $\kappa = \rho^2$, and let $H_\theta := A_\theta \rtimes_\ rho \mathbb Z_6$ and $C_\theta := A_\theta \rtimes_\kappa \mathbb Z_3$ denote the associated $C^*$-crossed products by corresponding cyclic groups. It is shown that for each $\theta$ there are canonical inclusions $\mathbb Z^{10} \hookrightarrow K_0(H_\theta)$ and $\mathbb Z^8 \hookrightarrow K_0(C_\theta)$ given explicitly by projections and ``mysterious'' modules (called {\it hexic} and {\it cubic} modules). We also find the unbounded traces on the canonical smooth dense $*$-subalgebras and so obtain Connes' cyclic cohomology groups of order zero $\text{HC}^0(H_\theta) \cong \mathbb C^{9}, \ \text{HC}^0(C_\theta) \cong \mathb b C^7$, when $\theta$ is irrational.

Keywords: Fourier transform, $C^*$-algebras, rotation algebras, automorphisms.

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