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

TL;DR

This paper computes the K-theory of crossed products of rotation algebras by free and amalgamated products of finite cyclic groups, revealing intricate behaviors of K-groups under various group actions and inclusions.

Contribution

It provides explicit calculations of K-groups for crossed products involving cyclic groups and explores the differences between free and amalgamated products.

Findings

K_0 groups are injective under certain inclusions, but not always.

K_1 vanishes for free products, but is non-zero for some amalgamated products.

The study uncovers non-trivial K-theory phenomena in these crossed products.

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 $A_\theta \to A_\theta \rtimes \mathbb Z_n$ induces injective maps on their $K_0$-groups, the same is not the case for the inclusions $A_\theta \rtimes \mathbb Z_d \to A_\theta \rtimes \mathbb Z_n$ for $2\le d < n \le 6$ and $d|n$, which we endeavor to calculate. Further, while for free products $K_1(A_\theta \rtimes [\mathbb Z_{m} \ast \mathbb Z_{n}]) = 0$, for amalgamated products $K_1(A_\theta \rtimes [\mathbb Z_{m} \ast_{\mathbb Z_{d}} \mathbb Z_{n}]) = \mathbb Z^k$ is non-vanishing ($k=1,2$).