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Definition

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

U ( f ) ( z ) = z f ( z ) V ( f ) ( z ) = f ( z e − 2 π i θ ) . {\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 θUV.¹

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.² 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 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 Aθ. The generating unitaries are the generator of the group Z and the identity function on the circle z : S1 → C.³
• 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 } {\displaystyle \{0\}} and itself.⁴
• Every irrational rotation algebra has a unique tracial state.⁵
• 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.⁶⁷

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.⁸

References

1. Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166, 218–219, 234. ISBN 0-8218-0599-1. 0-8218-0599-1 ↩

2. Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166, 218–219, 234. ISBN 0-8218-0599-1. 0-8218-0599-1 ↩

3. Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166, 218–219, 234. ISBN 0-8218-0599-1. 0-8218-0599-1 ↩

4. Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166, 218–219, 234. ISBN 0-8218-0599-1. 0-8218-0599-1 ↩

5. Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166, 218–219, 234. ISBN 0-8218-0599-1. 0-8218-0599-1 ↩

6. Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166, 218–219, 234. ISBN 0-8218-0599-1. 0-8218-0599-1 ↩

7. 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. /wiki/Marc_Rieffel ↩

8. 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. /wiki/Marc_Rieffel ↩