# 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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