Journal of Operator Theory

Volume 58, Issue 2, Fall 2007 pp. 441-462.

Non commutative spheres associated with the hexic transform and their K-theory

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, University of Northern British Columbia, Prince George, BC V2N 4Z9, Canada

Summary: Let

$A_\theta$ be the rotation

$C^*$ -algebra generated by unitaries

$U,V$ satisfying

$VU=\mathrm e^{2\pi \mathrm i\theta}UV$ and let

$\rho$ denote the hex ic transform on

$A_\theta$ defined by

$\rho(U)=V,\ \ \rho(V)=\mathrm e^{-\pi \mathrm i\theta} U^{-1}V$ . (It is the canonical order six automorphism of

$A_\theta$ .) It is shown that ten canonical classes in

$K_0(A_\theta\rtimes_\rho\mathbb Z_6) \cong \mathbb Z^{10}$ yield a basis. The Connes-Chern character

$K_0(A_\theta\rtimes_\rho\mathbb Z_6) \to H^{\rm{ev}}(A_\theta\rtimes_\rho\mathbb Z_6)^*$ is shown to be injective for each

$\theta$ , and its range is determined.

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