User:Mauritzvdworm/Noncommutative torus

The Noncommutative Torus, also known as the irrational rotation algebra, is one of the most fundamental examples of noncommutative geometry. It clearly highlights the key concepts regarding noncommutativity and opens the doors of this field of study to more complex noncommutative spaces.

Definition
Consider the space of square integrable functions on the torus denoted by $$L^2(\mathbb{T}^2)$$. It is known that this space is in fact a Hillbert space. We define the following operators

$$ Uf(x,y)=e^{2\pi ix}f\left(x,y+\frac{\theta}{2}\right) $$

$$ Vf(x,y)=e^{2\pi iy}f\left(x-\frac{\hbar}{2},y\right). $$

It can be shown that these operators are unitary and satisfy the commutation relation

$$ UV=e^{2\pi i\theta}VU. $$

The noncommutative torus is then defined as the $C^*$-algebra generated by the unitary operators $$U$$ and $$V$$.

The Noncommutative Torus as a Crossed Product
Using the $$C^*$$-dynamical system approach it can be shown that the noncommutative torus can be written as

$$A_{\theta}=C(\mathbb{T})\times_{\psi}\mathbb{Z}$$

where $$\psi$$ is the automorphism of $$C(\mathbb{T})$$ induced by a rotation of $$\theta$$. The case where $$\theta$$ is irrational is of special interest and in this case $$A_{\theta}$$ is a simple $$C^*$$-algebra.

The K-Theory of the Noncommutative Torus
In order to fully understand the $C^*$-algebra which is the quantum torus we need to determine its K-theory. The PV-sequence (Pimsner Voiculescu short exact sequence) is the main tool which we will be using to find the K-theory of the quantum torus. This sequence is also sometimes referred to as the six term exact sequence of K-theory.