COMMUNICATION A Case Study in Noncommutative Topology Claude L. Schochet Communicated by Gerald B. Folland space 푋 is compact Hausdorff and 퐴 ≅ 퐶(푋). So studying ABSTRACT. This is an expository note focused upon commutative unital 퐶∗-algebras is the same as studying ∗ one example, the irrational rotation 퐶 -algebra. We compact Hausdorff spaces—a natural category for alge- discuss how this algebra arises in nature—in quantum braic topology. Most 퐶∗-algebras are noncommutative, mechanics, group actions, and foliated spaces, and and so studying them we explain how 퐾-theory is used to get information is doing noncommuta- out of it. tive topology! Mathematicians Our goal is to write as if we are sitting in struggled to keep Introduction a coffeehouse and ex- up, to keep the This is the opposite of a survey paper. Here we are plaining an idea to a interested in one example, usually known as the irrational good friend (on nap- ∗ physics honest. rotation 퐶 -algebra or noncommutative torus and written kins, of course). So we 퐴휆, where 휆 is some irrational number between 0 and 1. are interested in get- We will show that 퐴휆 arises in at least three quite different ting an idea across but not at all interested in the contexts: technical details that, in any event, would be lost if the (1) quantum mechanics, coffee spilled.1 (2) action of a group on a compact Hausdorff space, (3) foliated spaces. Quantum Mechanics Then we will use 퐾-theory and traces to show that for 휆 In 1926–1927 the quantum-mechanical revolution in irrational between 0 and 1/2, the 퐴휆 are all nonisomorphic. physics changed our understanding of the world. As ∗ A word about the title. Commutative 퐶 -algebras with has been the pattern since, the physicists knew what they unit (“unital”) and compact Hausdorff spaces are equiva- wanted, and the mathematicians struggled to keep up, to lent categories: given a compact Hausdorff space 푋 you ∗ keep the physics honest (as a mathematician would put can form 퐶(푋), the commutative unital 퐶 -algebra of it). all continuous complex-valued functions on 푋, and given ∗ The simplest model of the hydrogen atom revolved a commutative unital 퐶 -algebra 퐴, its maximal ideal about two self-adjoint operators 푃 and 푄 that were to measure position and momentum of the electron. Claude (Chaim) Schochet is visiting professor of mathematics at the Technion, in Haifa, Israel, having retired emeritus from Heisenberg and Born showed that if 푄 is the position Wayne State University. His email address is [email protected]. operator and 푃 the momentum operator, then we have the canonical commutation relation It is a pleasure to thank Marc Rieffel who, besides contributing the most important theorems to this note, was very helpful in its 푃푄 − 푄푃 = −푖ℏ퐼, preparation, and to thank my old friend and co-author Jonathan where ℏ is Planck’s constant. Rosenberg. For permission to reprint this article, please contact: 1I learned this technique from Dror Bar-Natan, who gave a great [email protected]. colloquium talk entitled “From Stonehenge to Witten, Skipping all DOI: http://dx.doi.org/10.1090/noti1684 the Details.” June/July 2018 Notices of the AMS 653 COMMUNICATION It is easy to see that there are no 푛 × 푛 matrices 푃 and the norm-closed ∗-algebra generated by 푇 and by all of 푄 such that the 푀푓. Note that finite sums of the form 푃푄 − 푄푃 = −푖ℏ퐼 푘 푛 Σ푛=−푘푀푓푛 푇 with ℏ ≠ 0: just observe that are dense in 퐶(푆1) ⋊ ℤ. For 휆 irrational there is a unique 푡푟(푃푄 − 푄푃) = 푡푟(푃푄) − 푡푟(푄푃) = 0, normalized trace3 휏 on 퐶(푆1) ⋊ ℤ given on finite sums by but 푡푟(−ℏ퐼) = −푖푛ℏ. It is not much harder to see that 푛 ∫ there are no bounded self-adjoint operators on a Hilbert 휏(Σ푛푀푓푛 푇 ) = 푓0(푡)푑푡 ∈ ℝ 푆1 space with this property. There are unbounded ones, but where 푑푡 is normalized Lebesgue measure on the circle. to avoid technicalities with such operators it is best to It is not at all hard to prove that if 휆 is irrational, so that pass to the corresponding one-parameter unitary groups the action of ℤ on the circle is free, then 푖푠푃 푖푡푄 푈푠 = 푒 , 푉푡 = 푒 , 1 퐴휆 ≅ 퐶(푆 ) ⋊ ℤ. for which the commutation relation 푃푄 − 푄푃 = −푖ℏ becomes Foliated Spaces −푖ℏ푠푡 푈푠푉푡 = 푒 푉푡푈푠. The local picture of a foliated space is ℝ푝 × 푁, where 푁 is This is called the Weyl form of the canonical commutation some topological space. A subset of the form ℝ푝 × {푛} is relation. Such pairs of unitary groups exist: for example, called a plaque and a measurable subset 푇 ⊆ ℝ푝 ×푁 which 푖푡푥 2 푈푠푓(푥) = 푓(푥 − ℏ푠) and 푉푡푓(푥) = 푒 푓(푥) on 퐿 (ℝ). More- meets each plaque at most countably often (the simplest over, by the Stone-von Neumann theorem they are all being {푥} × 푁) is called a transversal. The global picture unitarily equivalent, subject to an irreducibility condition. is more complicated. We say that a (typically compact Taking 푠 = 푡 = 1 and 휆 = −2휋ℏ, we obtain unitary Hausdorff) space 푋 is a foliated space if each point in 푋 operators 푈 and 푉 such that has an open neighborhood homeomorphic to the local picture and locally the plaques fit together smoothly. 푈푉 = 푒2휋푖휆푉푈. A leaf is a maximal union of overlapping plaques; by These operators are bounded operators on the same construction it is a smooth 푝-dimensional manifold. Hilbert space, 푈, 푉 ∈ ℬ(ℋ). So we may take the (non- Here is an example. Start with the unit square [0, 1] × commuting) polynomial algebra generated by 푈, 푉, and [0, 1] foliated by the vertical lines {푥}×[0, 1]. Glue the left their adjoints. We then close up this algebra with respect to and right sides of the square together—that is, identify ∗ the operator norm and reach our goal, the 퐶 -algebra 퐴휆, (0, 푡) with (1, 푡)—to make a cylinder, again foliated by constructed visibly as a norm-closed, ∗-closed subalgebra vertical lines. If we now glue the top and bottom of the of 퐵(ℋ). cylinder together, identifying (푥, 0) with (푥, 1) we get a This is the first construction of the 퐴휆. We may restrict torus foliated by circles that go around the central hole. attention to 휆 ∈ [0, 1) and ask an elementary question: as (See Figure 1.) 휆 changes, how is 퐴휆 affected? It turns out that the case of greatest interest is when 휆 is irrational, and so we will restrict to that case as needed. Homeomorphisms of the Circle Let 휙 ∶ 푆1 → 푆1 be rotation of the circle by 2휋휆 radians counterclockwise. Any rotation is a homeomorphism and thus determines an action of the integers on the circle by sending 푛 to 휙푛. This defines an action of the integers on 퐶(푆1) and from this one can construct a 퐶∗-algebra 퐶(푆1) ⋊ ℤ as follows.2 For 휆 = 0 or 1, 퐶(푆1) ⋊ ℤ is simply 퐶(푇2), continuous functions on the torus. For 휆 irrational we can realize this algebra as follows. Take ℋ to be the Hilbert space 퐿2(푆1) and let 푇 ∈ ℬ(ℋ) be the bounded invertible operator 푇푓 = 푓 ∘ 휙. Any 푓 ∈ 퐶(푆1) gives a pointwise 1 multiplication operator 푀푓 ∈ ℬ(ℋ). Then 퐶(푆 ) ⋊ ℤ is Figure 1. A torus foliated by lines of given irrational slope gives rise to a 퐶∗-algebra. 2This is actually a quite general construction. Starting with any 퐶∗-algebra and locally compact group 퐺 acting on 퐴 you get To get something more interesting, we can give the a new crossed-product 퐶∗ algebra built by taking the algebraic cylinder a twist before gluing the ends together: we glue crossed product and then completing. There are some topologi- cal choices for how the completion is done, but if the group is 3A normalized trace on 퐴 is a linear functional on 퐴 such that commutative then the completions coincide. 휏(푥∗푥) = 휏(푥푥∗) ≥ 0 for all 푥 and 휏(1) = 1. 654 Notices of the AMS Volume 65, Number 6 COMMUNICATION (푥, 0) to (푥 ⊕ 휆, 1), where “⊕” means addition mod 1. Now by elementary considerations. But what is the range of something quite unusual happens, and here we must the map 휏 ? In the case 퐴 = 푀푛(ℂ) the range of 휏 would 1 2 specify whether 휆 is rational or not. If 휆 is rational, then be the set {0, 푛 , 푛 , … , 1} ⊆ ℝ. What happens for 퐴휆? Stay each leaf of the foliated space is actually a circle wrapped tuned. ∗ around the torus several times. On the other hand, if 휆 is To summarize, we have shown that the 퐶 -algebra 퐴휆 irrational, then you do not get circles: every leaf is a line arises in three disparate arenas of mathematics. (There ℝ. Furthermore, as in Figure 1, each line is wrapped about are others as well, but this should be enough to convince the torus infinitely often: each line is actually dense in the you that it happens a lot!) At this point, though, it is not torus. This construction is called the Kronecker flow on at all clear to what extent the algebra is dependent upon the torus. 휆. Let’s find out. Every foliated space satisfying very mini- The World’s Fastest Introduction to 퐾-theory mal technical assump- There is a natural Suppose first that 퐴 is a unital 퐶∗-algebra.