Noncommutative torus

<h2 id="definition">Definition</h2>
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 <a href="/facts/Bounded_linear_operator/LYmDTIqR">bounded linear operators</a> of <a href="/facts/Square-integrable_function/IASblcJJ">square-integrable functions</a> on the <a href="/facts/Unit_circle/VsTffjPK">unit circle</a> 
 
 
 
 
 S
 
 1
 
 
 ⊂
 
 C
 
 
 
 {\displaystyle S^{1}\subset \mathbb {C} }
 
, generated by two <a href="/facts/Unitary_operator/rkypfQQ9">unitary</a> operators 
 
 
 
 U
 ,
 V
 
 
 {\displaystyle U,V}
 
 defined as<blockquote>
 
 
 
 
 
 
 
 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}}}
 
</blockquote>A quick calculation shows that VU = e−2π i θUV.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
<h2 id="alternative-characterizations">Alternative characterizations</h2>
<ul><li>Universal property: Aθ can be defined (up to isomorphism) as the <a href="/facts/Universal_C*-algebra/7WKHf66p">universal C*-algebra</a> generated by two unitary elements U and V satisfying the relation VU = e2π i θUV.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> This definition extends to the case when θ is rational. In particular when θ = 0, Aθ is isomorphic to continuous functions on the <a href="/facts/Torus/J9SkaV1W">2-torus</a> by the <a href="/facts/Gelfand_representation/QV54f7Jm">Gelfand transform</a>.</li>
<li>Irrational rotation algebra: Let the infinite cyclic group Z act on the circle S1 by the <a href="/facts/Irrational_rotation/1xQtFceA">rotation action</a> by angle 2πiθ. This induces an action of Z by automorphisms on the algebra of continuous functions C(S1). The resulting C*-<a href="/facts/Crossed_product/AVtwntCg">crossed product</a> 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.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li>
<li>Twisted group algebra: The function σ : Z2 × Z2 → C; σ((m,n), (p,q)) = e2πinpθ is a <a href="/facts/Group_cohomology/ncLTIHj4">group 2-cocycle</a> on Z2, and the corresponding twisted <a href="/facts/Group_algebra_of_a_locally_compact_group/I1BVHqYD">group algebra</a> C*(Z2; σ) is isomorphic to Aθ.</li></ul>
<h2 id="properties">Properties</h2>
<ul><li>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.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a></li>
<li>Every irrational rotation algebra has a unique <a href="/facts/State_(functional_analysis)/HR8Vvjvg">tracial state</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li>
<li>The irrational rotation algebras are <a href="/facts/Nuclear_C*-algebra/bMBywnFW">nuclear</a>.</li></ul>
<h2 id="classification-and-k-theory">Classification and K-theory</h2>
The <a href="/facts/Operator_K-theory/GkTW22Ur">K-theory</a> of Aθ is Z2 in both even dimension and odd dimension, and so does not distinguish the irrational rotation algebras. But as an <a href="/facts/Partially_ordered_group/xoLNpEqH">ordered group</a>, 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.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
Two irrational rotation algebras Aθ and Aη are <a href="/facts/Morita_equivalence/vEVRJSJw">strongly Morita equivalent</a> if and only if θ and η are in the same orbit of the action of SL(2, Z) on R by <a href="/facts/Fractional_linear_transformations/nykA54lI">fractional linear transformations</a>. 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.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166, 218–219, 234. ISBN 0-8218-0599-1. <a href="0-8218-0599-1" target="_blank">0-8218-0599-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166, 218–219, 234. ISBN 0-8218-0599-1. <a href="0-8218-0599-1" target="_blank">0-8218-0599-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166, 218–219, 234. ISBN 0-8218-0599-1. <a href="0-8218-0599-1" target="_blank">0-8218-0599-1</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166, 218–219, 234. ISBN 0-8218-0599-1. <a href="0-8218-0599-1" target="_blank">0-8218-0599-1</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166, 218–219, 234. ISBN 0-8218-0599-1. <a href="0-8218-0599-1" target="_blank">0-8218-0599-1</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166, 218–219, 234. ISBN 0-8218-0599-1. <a href="0-8218-0599-1" target="_blank">0-8218-0599-1</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Marc_Rieffel" target="_blank">/wiki/Marc_Rieffel</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Marc_Rieffel" target="_blank">/wiki/Marc_Rieffel</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Noncommutative torus open-in-new

Noncommutative torus