Noncommutative torus

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In mathematics, and more specifically in the theory of C*-algebras, the noncommutative tori Aθ (also known as irrational rotation algebras when θ is irrational) are 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[edit]

For any irrational number θ, the noncommutative torus Aθ is the C*-subalgebra of B(L^2(\mathbb{T})), the algebra of bounded linear operators of square-integrable functions on the unit circle of \mathbb{C} generated by unitary elements U and V, where U(f)(z)=zf(z) and V(f)(z)=f(e^{-2\pi i\theta}z). A quick calculation shows that VU = e^{-2\pi i \theta}UV.[1]

Alternative characterizations[edit]

  • 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 = e^{2\pi i \theta}UV.[1] 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 : S1C.[1]
  • Twisted group algebra: The function σ : Z2 × Z2C; σ((m,n), (p,q)) = einpθ is a group 2-cocycle on Z2, and the corresponding twisted group algebra C*(Z2; σ) is isomorphic to Aθ.

Classification and K-theory[edit]

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, K0Z + θZ. Therefore two noncommutative tori A_{\theta} and A_{\eta} are isomorphic if and only if either θ+η or θ-η is an integer.[1][2]

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.[2]

References[edit]

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