Noncommutative torus


In mathematics, and more specifically in the theory of C*-algebras, the noncommutative tori Aθ, also known as irrational rotation algebras for irrational values of θ, form a family of noncommutative C*-algebras which generalize the algebra of continuous functions on the 2-torus. Many topological and geometric properties of the classical 2-torus have algebraic analogues for the noncommutative tori, and as such they are fundamental examples of a noncommutative space in the sense of Alain Connes.

Definition

For any real number θ, the noncommutative torus Aθ is the C*-subalgebra of B, the algebra of bounded linear operators of square-integrable functions on the unit circle S1 of C, generated by the unitary elements U and V, where U=zf and V=f. A quick calculation shows that VU = e−2πiθUV.

Alternative characterizations

The K-theory of Aθ is Z2 in both even dimension and odd dimension, and so does not distinguish the irrational rotation algebras. But as an ordered group, K0 ≃ Z + θZ. Therefore, two noncommutative tori Aθ and Aη are isomorphic if and only if either θ + η or θ − η is an integer.
Two irrational rotation algebras Aθ and Aη are strongly Morita equivalent if and only if θ and η are in the same orbit of the action of SL on R by fractional linear transformations. In particular, the noncommutative tori with θ rational are Morita equivalent to the classical torus. On the other hand, the noncommutative tori with θ irrational are simple C*-algebras.