Noncommutative torus explained

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

A _{\theta} is the C*-subalgebra of B(L ^{2(R/Z))} , the algebra of bounded linear operators of square-integrable functions on the unit circle S ^{1} \subset 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} ). \end{align}

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. 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 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 A_θ. 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)) = 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

\{ 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 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.^[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.

Notes and References

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