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 irrational real number θ, the noncommutative torus ${\displaystyle A_{\theta }}$ is the C*-subalgebra of ${\displaystyle B(L^{2}(\mathbb {R} /\mathbb {Z} ))}$, the algebra of bounded linear operators of square-integrable functions on the unit circle ${\displaystyle S^{1}\subset \mathbb {C} }$, generated by two unitary operators ${\displaystyle U,V}$ defined as

${\displaystyle {\begin{aligned}U(f)(z)&=zf(z)\\V(f)(z)&=f(ze^{-2\pi i\theta }).\end{aligned}}}$

A quick calculation shows that VU = e^{−2π i θ}UV.^[1]

Alternative characterizations

• Universal property: A_θ can be defined (up to isomorphism) as the universal C*-algebra generated by two unitary elements U and V satisfying the relation VU = e^{2π i θ}UV.^[1] This definition extends to the case when θ is rational. In particular when θ = 0, A_θ 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 2πiθ. 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 A_θ. The generating unitaries are the generator of the group Z and the identity function on the circle z : S¹ → C.^[1]
• Twisted group algebra: The function σ : Z² × Z² → C; σ((m,n), (p,q)) = e^2πinpθ is a group 2-cocycle on Z², and the corresponding twisted group algebra C*(Z²; σ) is isomorphic to A_θ.

Properties

• Every irrational rotation algebra A_θ is simple, that is, it does not contain any proper closed two-sided ideals other than ${\displaystyle \{0\}}$ and itself.^[1]
• Every irrational rotation algebra has a unique tracial state.^[1]
• The irrational rotation algebras are nuclear.

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, K₀ ≃ Z + θZ. Therefore, two noncommutative tori A_θ and A_η are isomorphic if and only if either θ + η or θ − η is an integer.^[1][2]

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.^[2]

References

1. Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166, 218–219, 234. ISBN 0-8218-0599-1.
2. Rieffel, Marc A. (1981). "C*-Algebras Associated with Irrational Rotations" (PDF). Pacific Journal of Mathematics. 93 (2): 415–429 [416]. doi:10.2140/pjm.1981.93.415. Retrieved 28 February 2013.