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.
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Definition
For any real number θ, the noncommutative torus Aθ is the C*-subalgebra of B(L2(T)), the algebra of bounded linear operators of square-integrable functions on the unit circle of C generated by unitary elements U and V, where U(f)(z)=zf(z) and V(f)(z)=f(e−2π iθz). A quick calculation shows that VU = e−2πiθUV.
Alternative characterizations
Classification and K-theory
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 ordered groups, 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(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.