K-theory of Rotation Algebra Crossed Products by Amalgamated Products of Finite Cyclic Groups

Abstract

The $K$-groups of the crossed product of the rotation C*-algebra $A_{\theta }$ by free and amalgamated products of the cyclic groups ${\mathbb{Z}}_n$, for $n=2,3,4,6$, are calculated. The actions here arise from the canonical actions of these groups on the rotation algebra under the flip, cubic, Fourier, and hexic automorphisms, respectively. An interesting feature in this study is that although the inclusion $A_{\theta }\to A_{\theta }⋊{\mathbb{Z}}_n$ induces injective maps on their $K_0$-groups, the same is not the case for the inclusions $A_{\theta }⋊{\mathbb{Z}}_d\to A_{\theta }⋊{\mathbb{Z}}_n$ for $2\le d<n\le 6$ and $d|n$, which we endeavor to calculate. Further, while for free products $K_1(A_{\theta }⋊[{\mathbb{Z}}_m\ast {\mathbb{Z}}_n])=0$, for amalgamated products $K_1(A_{\theta }⋊[{\mathbb{Z}}_m{\ast }_{{\mathbb{Z}}_d}{\mathbb{Z}}_n])={\mathbb{Z}}^k$ is non-vanishing ($k=1,2$).

16 pages

References in corpus (1)

• The AF structure of non commutative toroidal Z/4Z orbifolds