Noncommutative torus

For any irrational real number θ, the noncommutative torus A_\theta is the C*-subalgebra of B(L^2(\mathbb{R}/\mathbb{Z})), the algebra of bounded linear operators on square-integrable functions on the unit circle S^1 \subset \mathbb{C}, generated by two unitary operators U, V defined as\begin{align} U(f)(z) &= z f(z) \\ V(f)(z) &= f(ze^{-2\pi i \theta}). \end{align}A quick calculation shows that VU = e−2π i θUV. {{cite book == Alternative characterizations ==

• Universal property: Aθ can be defined (up to isomorphism) as the universal C*-algebra generated by two unitary elements U and V satisfying the relation VU = e2π i θUV. This definition extends to the case when θ is rational. In particular when θ = 0, Aθ is isomorphic to continuous functions on the 2-torus by the Gelfand transform. • Irrational rotation algebra: Let the infinite cyclic group Z act on the circle S1 by the rotation action by angle 2iθ. This induces an action of Z by automorphisms on the algebra of continuous functions C(S1). The resulting C*-crossed product C(S1) ⋊ Z is isomorphic to Aθ. The generating unitaries are the generator of the group Z and the identity function on the circle z : S1 → C. • Twisted group algebra: The function σ : Z2 × Z2 → C; σ((m,n), (p,q)) = e2πinpθ is a group 2-cocycle on Z2, and the corresponding twisted group algebra C*(Z2; σ) is isomorphic to Aθ. ==Properties==

• Every irrational rotation algebra Aθ is simple, that is, it does not contain any proper closed two-sided ideals other than \{0\} and itself. • Every irrational rotation algebra has a unique tracial state. • The irrational rotation algebras are nuclear. == 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 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(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. == References ==