en>MBD123: grammar

2013-05-26T01:28:47Z

grammar

New page

In [[mathematics]], and more specifically in the theory of [[C*-algebra]]s, the '''noncommutative tori''' ''A''θ (also known as '''irrational rotation algebras''' when θ is irrational) are a family of noncommutative C*-algebras which generalize the [[C*-algebra#Commutative_C.2A-algebras|algebra of continuous functions]] on the [[Torus|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 geometry|noncommutative space]] in the sense of [[Alain Connes]].

== Definition ==
For any irrational number θ, the noncommutative torus ''A''θ is the C*-subalgebra of <math>B(L^2(\mathbb{T}))</math>, the algebra of bounded linear operators of square-integrable functions on the unit circle of <math>\mathbb{C}</math> generated by unitary elements <math>U</math> and <math>V</math>, where <math>U(f)(z)=zf(z)</math> and <math>V(f)(z)=f(e^{-2\pi i\theta}z)</math>. A quick calculation shows that <math>VU = e^{-2\pi i \theta}UV</math>.<ref name="Davidson97">
{{cite book
|last=Davidson |first=Kenneth
|title=C*-Algebras by Example
|year=1997
|publisher=Fields Institute
|isbn=0-8218-0599-1
|pages=166,218–219,234}}</ref>

== 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 <math>VU = e^{2\pi i \theta}UV.</math><ref name="Davidson97" /> This definition extends to the case when θ is rational. In particular when θ=0, ''A''θ is isomorphic to continuous functions on the [[torus|2-torus]] by the [[Gelfand_representation#The_C.2A-algebra_case|Gelfand transform]].
* '''Irrational rotation algebra:''' Let the infinite cyclic group '''Z''' act on the circle '''S'''1 by the [[irrational rotation|rotation action]] by angle 2π''i''θ. This induces an action of '''Z''' by automorphisms on the algebra of continuous functions ''C''('''S'''1). The resulting C*-[[crossed product]] ''C''('''S'''1) ⋊ '''Z''' is isomorphic to ''A''θ. The generating unitaries are the generator of the group '''Z''' and the identity function on the circle ''z'' : '''S'''1 → '''C'''.<ref name="Davidson97" />
* '''Twisted group algebra:''' The function σ : '''Z'''2 × '''Z'''2 → '''C'''; σ((''m'',''n''), (''p'',''q'')) = ''e''2π''inp''θ is a [[group cohomology|group 2-cocycle]] on '''Z'''2, and the corresponding twisted group algebra ''C*''('''Z'''2; σ) is isomorphic to ''A''θ.

== Classification and K-theory ==
The [[Operator K-theory|K-theory]] of ''A''θ is '''Z'''2 in even dimension and trivial in odd dimension, and so does not distinguish the irrational rotation algebras. But as ordered groups, ''K''0 ≃ '''Z''' + θ'''Z'''. Therefore two noncommutative tori <math>A_{\theta}</math> and <math>A_{\eta}</math> are isomorphic if and only if either θ+η or θ-η is an integer.<ref name=Davidson97 /><ref name=Rieffel81>{{cite journal|last=Rieffel|first=Marc A.|title=C*-Algebras Associated with Irrational Rotations|journal=Pacific Journal of Mathematics|year=1981|volume=93|issue=2|doi=10.2140/pjm.1981.93.415|pages=415–429 [416]|url=http://msp.org/pjm/1981/93-2/pjm-v93-n2-p12-s.pdf|accessdate=28 February 2013}}</ref>

Two irrational rotation algebras ''A''θ and ''A''η are [[Morita equivalence#Further_directions|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.<ref name=Rieffel81 />

== References ==
<references/>

[[Category:C*-algebras]]

Multipole magnet - Revision history

en>MBD123: grammar