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

$\begin U(f)(z) &= e^f(z) \\ V(f)(z) &= f(z - \theta) \end$

where $e^$ is the parametrization of the point $z$ on the circle $\mathbb/\mathbb$ in $\mathbb$. A quick calculation shows that ''VU'' = ''e''^{−2π ''i'' θ}''UV''.

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 2''iθ''. 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''^2π''inpθ'' 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 $\$ 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, ''K''₀ ≃ Z + ''θ''Z. Therefore, two noncommutative tori ''A''_''θ'' and ''A''_''η'' are isomorphic if and only if either ''θ'' + ''η'' or ''θ'' − ''η'' is an integer.

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.

References

{{Reflist

C*-algebras
Noncommutative geometry