A Case Study in Noncommutative Topology

COMMUNICATION

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 topological 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. There are tions has a 퐶∗-algebra trace that arises. always projections, namely 0 and 1. If 푋 is a connected associated to it. This space then these are the only projections in 퐶(푋). On the is due to H. E. other hand, 푀푛(ℂ) has lots of projections: for instance, Winkelnkemper (1983) and to A. Connes (1979). In our take a diagonal matrix that has only ones and zeros on context these always have the form 퐴 ⊗ 풦 where 풦 is the diagonal. It turns out that 퐶(푋) ⊗ 푀푛(ℂ) can have the 퐶∗-algebra of the compact operators.4 Here are some very interesting projections—these correspond to vector examples: bundles over 푋. (1) If 푋 = 퐹×퐵 or more generally, if 푋 is the total space Let 푃푛(퐴) denote the set of projections in 퐴 ⊗ 푀푛(ℂ), of a compact fibre bundle with fibre some smooth and define 푃∞(퐴) to be the union of the 푃푛(퐴) (where manifold 퐹 and base the compact Hausdorff space we put 푃푛 inside of 푃푛+1 by sticking it in the upper left 퐵, then 푋 is foliated by the fibres and the foliation corner and adding zeros to the right and below). Unitary algebra is 퐶(퐵) ⊗ 풦. equivalence and saying that 푝 is equivalent to 푝⊕0 puts a (2) If 푋 is the torus foliated by circles, then it is a natural equivalence relation ∼ on 푃∞(퐴). Then 푃∞(퐴)/ ∼ fibre bundle of the form 푆1 → 푋 → 푆1 and by (1) has a natural direct sum operation, and we can turn it the foliation algebra is 퐶(푆1) ⊗ 풦 (where the 푆1 in into an abelian group by doing the so-called Grothendieck 퐶(푆1) is the base of the fibre bundle, not the generic construction (taking formal differences of projections). If fibre). you don’t like that, take the free abelian group on the (3) (the punch line) If 푋 is the Kronecker flow on the equivalence classes and then divide out by the subgroup generated by all elements of the form [푃⊕푄]−[푃]−[푄]. torus, then the foliation algebra is 퐴휆 ⊗ 풦. This gives an abelian group denoted 퐾 (퐴). Note for 0 < 휆 < 1/2 that using 휆 or 1 − 휆 gives the 0 For example, take 퐴 = ℂ. Then 푃 (퐴) consists of same foliated space, and so 퐴 ≅ 퐴 . Thus, we restrict 푛 휆 1−휆 all of the projections in 푀 (ℂ). We learned in the sec- attention to irrational 휆 between 0 and 1/2. 푛 ond semester of linear algebra that every projection is There is a natural trace that arises in this construction unitarily equivalent to a diagonal matrix of the form as well. What is needed is an invariant transverse measure. 푑푖푎푔(1, … , 1, 0, … , 0). Hence, the equivalence classes of A transverse measure measures transversals, naturally 푃 (ℂ) are classified (via rank) by the integers {0, 1, 2, … , 푛}, enough. If it has enough nice properties then it is an 푛 and the equivalence classes of 푃 (ℂ) are classified by the invariant transverse measure. Not all foliated spaces have ∞ natural numbers {0, 1, 2, … }. Taking formal inverses, we them, but the ones we are looking at do. In the case of obtain 퐾 (ℂ) ≅ ℤ. Note that the same answer emerges if the fibre bundle above, the foliation algebra is simply 0 we take 퐴 = 푀 (ℂ) for any 푗, since “matrices of matrices 퐶(퐵) ⊗ 풦 and invariant transverse measures correspond 푗 are matrices.” to certain measures on 퐵. Invariant transverse measures For commutative unital 퐶∗-algebras 퐴 = 퐶(푋) with correspond to Ruelle-Sullivan currents in foliation theory compact Hausdorff maximal ideal space 푋, the Serre-Swan (cf. [1] Ch. IV). theorem tells us that 퐾 (퐶(푋)) is given by In the case of the Kronecker flow on the torus, the 0 0 invariant transverse measure may be constructed from 퐾0(퐶(푋)) ≅ 퐾 (푋) Lebesgue measure on a transverse circle to the foliated where 퐾0(푋) is the Grothendieck group generated by space. This passes to a trace on the foliation algebra complex vector bundles over 푋. ∗ which corresponds to the trace constructed above. We may regard 퐾0 as a functor on unital 퐶 -algebras 2 A projection 푝 is an element of 퐴 that satisfies 푝 = 푝 = and maps, since if 푓 ∶ 퐴 → 퐴′ is unital, then 푓 takes ∗ 푝 . Suppose that 푝 ∈ 퐴 is a projection. Then 0 ≤ 휏(푝) ≤ 1 projections to projections, unitaries to unitaries, and preserves direct sums. If 퐴 is not unital, then we may 4 If you are an analyst you can think of 풦 as the smallest form its unitization 퐴+ (for example, 퐶 (푋)+ ≅ 퐶(푋+) ∗ 표 퐶 -algebra inside the space of bounded operators ℬ(ℋ) on where 푋 is locally compact but not compact and 푋+ is its a separable Hilbert space ℋ that contains all of the operators one-point compactification), and then define 퐾 (퐴) to be with finite-dimensional range. If you are an algebraist at heart 0 the kernel of the map then you will be pleased to hear that 풦 is the unique maximal + + ideal of ℬ(ℋ). 퐾0(퐴 ) ⟶ 퐾0(퐴 /퐴) ≅ ℤ.

June/July 2018 Notices of the AMS 655 COMMUNICATION

Since we are generally working with algebras tensored (3) More generally, if 휆 and 휇 are irrational numbers in with the compact operators 풦, it is good to know that the interval [0, 1/2] and 푟 and 푠 are positive integers 퐾0(퐴) ≅ 퐾0(퐴 ⊗ 풦). with Note that if 퐴 is separable then there are at most 퐴휆 ⊗ 푀푟(ℂ) ≅ 퐴휇 ⊗ 푀푠(ℂ) countably many equivalence classes of projections, and then 휆 = 휇 and 푟 = 푠. hence 퐾0(퐴) is a countable abelian group. Every countable (4) Suppose that both 휆 and 휇 are irrational. Then 퐴휆 ⊗ abelian group may be realized as 퐾0(퐴) for some separable 풦 ≅ 퐴휇 ⊗ 풦 if and only if 휆 and 휇 are in the same ∗ 퐶 -algebra 퐴. orbit of the action of 퐺퐿(2, ℤ) on irrational numbers by linear fractional transformations. The 퐾-theory of the Irrational Rotation 퐶∗-algebra: The Bad News So we see that the 퐴휆 retain all of the sensitive information about the angle 휆. If we think back to the Now, what is the 퐾-theory of the irrational rotation origins of 퐴 this seems really astonishing: The angle 퐶∗-algebra? A seemingly elementary question arises first: 휆 of the Kronecker flow deeply affects the geometry of the does 퐴 have any non-trivial projections? This was open 휆 foliated space. for several years, and it led to decisive work by Marc Rieffel whose results, together with those of Pimsner and References Voiculescu, we now describe.5 [1] C. C. Moore and C. Schochet, Global Analysis on Foliated We are altering the historical order a bit in what follows. Spaces, Second edition, Mathematical Sciences Research Insti- If 휆 is irrational then Pimsner and Voiculescu (1980) tute Publications, 9, Cambridge University Press, New York, showed that 2006. MR2202625 퐾0(퐴휆) ≅ ℤ ⊕ ℤ [2] M. A. Rieffel, 퐶∗-algebras associated with irrational independent of 휆. This is just like for 퐴 = 퐶(푇2)! So using rotations, Pacific J. Math. 93 (1981), 415–429. MR623572 퐾0 by itself we cannot distinguish the various 퐴휆. Photo Credits Traces to the Rescue Figure 1 courtesy of Andrea Gambassi and Corinna Ulcigrai. ∗ The key to distinguishing the family of 퐶 -algebras {퐴휆} Author photo courtesy of Rivka Schochet. is the trace defined at the end of the Homeomorphisms of the Circle section. If 퐴 is a 퐶∗-algebra with a normalized trace 휏 and 푝 and 푞 are orthogonal projections in 퐴 (i.e. 푝푞 = 0) then ABOUT THE AUTHOR 휏(푝 ⊕ 푞) = 휏(푝) + 휏(푞), Claude (Chaim) L. Schochet mostly works at home in the and so the trace gives us a homomorphism tiny village of Bar Yochai, Israel 휏 ∶ 퐾0(퐴) ⟶ ℝ looking out at Mt. Meron when of abelian groups. We have remarked previously that if 퐴 he is not visiting local wineries or playing folk music on his guitar. is separable then 퐾0(퐴) is a countable abelian group, and We welcome visitors! hence 휏(퐾0(퐴휆)), the image of Claude L. Schochet 휏 ∶ 퐾0(퐴휆) ⟶ ℝ, is a countable subgroup of ℝ. Pimsner and Voiculescu showed that the range of the trace 휏 퐾0(퐴휆) ≅ ℤ ⊕ ℤ ⟶ ℝ lies inside ℤ+휆ℤ, the subgroup of ℝ generated by 1 and by 휆. Rieffel showed that every element of (ℤ + 휆ℤ) ∩ [0, 1] is the image of a projection in 퐴휆. Combining these results gives us this omnibus isomorphism theorem: Theorem 1 (Rieffel 1981 [2], Pimsner and Voiculescu 1980). (1) If 휆 is irrational then the image of the trace

휏 ∶ 퐾0(퐴휆) ⟶ ℝ is exactly ℤ + 휆ℤ. (2) If 휆 and 휇 are irrational numbers in the interval [0, 1/2] with 퐴휆 ≅ 퐴휇 then 휆 = 휇.

5 It is easy to show that 퐴휆 ⊗ 풦 has projections, and those projections determine the 퐾-theory. It is a much deeper problem to deal with 퐴휆 itself.

656 Notices of the AMS Volume 65, Number 6