# A note on crossed products of rotation algebras

## TL;DR

This paper computes the K-theory of crossed products of rotation algebras by infinite order SL(2,Z) matrices, providing explicit generators and tracial ranges, advancing understanding of their algebraic structure.

## Contribution

It explicitly determines the K-theory, generators, and tracial ranges of crossed products of rotation algebras by certain matrices, using continuous field techniques.

## Key findings

- Computed K-theory for all real θ
- Identified explicit generators for K_0-groups
- Calculated tracial ranges concretely

## Abstract

We compute the $K$-theory of crossed products of rotation algebras $\mathcal{A}_\theta$, for any real angle $\theta$, by matrices in $\mathrm{SL}(2,\mathbb{Z})$ with infinite order. Using techniques of continuous fields, we show that the canonical inclusion of $\mathcal{A}_\theta$ into the crossed products is injective at the level of $K_0$-groups. We then give an explicit set of generators for the $K_0$-groups and compute the tracial ranges concretely.