Noncommutative torus

In mathematics, and more specifically in the theory of C*-algebras, the noncommutative tori "A"θ, 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 formula_1 is the C*-subalgebra of formula_2, the algebra of bounded linear operators of square-integrable functions on the unit circle formula_3, generated by two unitary operators formula_4formula_5where formula_6 is the parameterization of the circle formula_7 in formula_8. A quick calculation shows that "VU" = "e"−2π"i"θ"UV".
Classification and K-theory.
The K-theory of "A""θ" is Z2 in both even dimension and odd dimension, and so does not distinguish the irrational rotation algebras. But as an ordered group, "K"0 ≃ Z + "θ"Z. Therefore, two noncommutative tori "A""θ" and "A""η" are isomorphic if and only if either "θ" + "η" or "θ" − "η" is an integer.
Two irrational rotation algebras "A""θ" and "A""η" 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.