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.
For any irrational real number θ, the noncommutative torus
, the algebra of bounded linear operators of square-integrable functions on the unit circle
[1][2] 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(2, Z) 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.