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.

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