Quantum Spaces and Their Noncommutative Topology, Volume

Home , Noncommutative geometry, Noncommutative topology

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Quantum Spaces and Their Noncommutative Topology Joachim Cuntz

oncommutative geometry studies the differential operators, and the algebra of differential geometry of “quantum spaces”. Put a forms all on the same footing. little more prosaically, this means the Now, what is a “geometric” property of a non- “geometric properties” of noncommu- commutative algebra? How can one describe Ntative algebras (say, over the field C characteristic classes or additional structures of complex numbers). Such algebras include, for like a Riemannian metric for a noncommutative instance, algebra? These questions are what noncommuta- • algebras of pseudodifferential operators, tive geometry is all about; see the fascinating book algebras of leafwise differential operators on by Connes [5]. foliated manifolds, algebras of differential The two fundamental “machines” of noncom- forms, group algebras or convolution algebras mutative geometry are cyclic homology and for groupoids; (bivariant) topological K-theory. Cyclic theory can • noncommutative or “quantized” versions of be viewed as a far-reaching generalization of familiar algebras such as algebras of functions the classical de Rham cohomology, while bivariant on spheres, on tori, on simplicial complexes, K-theory includes the topological K-theory of or on classifying spaces; Atiyah-Hirzebruch as a very special case. genuinely new noncommutative algebras, Bivariant K-theory was first defined and devel- • ∗ for instance, ones motivated by quantum oped by Kasparov on the category of C -algebras, mechanics. thereby unifying and decisively extending previous The underlying philosophy is based on the ob- work by Atiyah-Hirzebruch, Brown-Douglas- servation that various categories of spaces can be Fillmore, and others. Kasparov also applied his completely described by the (commutative) algebras bivariant theory to obtain striking positive results of functions on them (a locally compact space by on the Novikov conjecture. Very recently, it was the algebra of continuous functions, a smooth discovered that bivariant topological K-theories manifold by the algebra of smooth functions, an can be defined on a wide variety of topological affine algebraic variety by its coordinate ring). The algebras ranging from discrete algebras and very general locally convex algebras to Banach algebras idea then is that a noncommutative algebra can or C∗-algebras. be viewed as an algebra of functions on a virtual Cyclic theory is a homology theory developed “noncommutative space”. This approach is very independently by Connes and by Tsygan, who were flexible: for instance, it covers the algebra of motivated by different aspects of K-theoretic con- functions on a manifold, the algebra of pseudo- structions. It was immediately realized that cyclic Joachim Cuntz is professor of mathematics at the homology has close connections with de Rham Universität Münster, Germany. His e-mail address theory, Lie algebra homology, group cohomology, is [email protected]. and index theorems.

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It is important to note that the new theories are possible paths in the following graph. It by no means simply generalizations of classical possesses, besides the 0-dimensional class constructions. In fact, in the commutative case given by the equivalence class of the points, a they provide a new approach and a quite unex- one-dimensional class coming from the path pected interpretation of the well-known classical around the circle (contrary to the case of theories. Essential properties of the two theories Mn(C) , we assume here that this path is become visible only in the noncommutative cate- different from the trivial path). gory. For instance, both theories have certain universality properties in this setting. Let us have a look at the kind of geometric information that the two theories give us for a number of simple “quantum spaces”. The formal definition of the cyclic and K-theory classes mentioned in these examples will be explained in the subsequent section. The technical definition is not necessary for an intuitive grasp of the situation. 1. The space with n points and noncommutative connections. This space has n points and arrows between every two points. As an algebra it is described by the algebra Mn(C) of n × n-matrices (the functions on the n points corresponding to the diagonal matrices). Both

2. The phase space in quantum mechanics. This is described by the unital algebra A(p, q) generated by two generators p and q satisfying the Heisenberg relation pq − qp =1 (sometimes this is called the Weyl algebra). There is at present no calculation of the K-theory for this algebra (or its C∞-completion). The cyclic theory sees one two-dimensional cohomology class and no classes in dimensions different from 2. Thus we have here a noncommutative space that is two-dimensional (say, looks like a 2-plane). However, not only does this “space” have no points, it does not even have any K-theory and cyclic theory see one even co- equivalence class of a point. homology class and no odd classes. In both theories the nontrivial even class is 0-dimensional, and there are no higher cohomology classes. Since there is one class representing the equivalence class of the n points and no higher dimensional classes, Mn(C) looks like a connected 0-dimensional space. This is the simplest case of a convolution algebra for a groupoid. In general a (topological) groupoid consists of a space of objects and a family of (invertible) arrows which can be considered as above as noncommutative paths between the objects. For the noncommutative homology theories, different points connected by an arrow will be homologous. Higher homology classes can also arise from configurations of arrows (like a loop of arrows), from mixed configurations involving arrows and objects, or even from linear combinations of such things. Consider, for instance, the alge- 3. The noncommutative 2-torus. This is the bra determined by linear combinations of all involutive unital algebra Aθ given by power

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series with rapidly decreasing coefficients dimension of a cyclic cohomology class for CΣ in two generators u and v satisfying the rela- is, however, much larger than the dimension tions uu∗ = u∗u = vv∗ = v∗v =1 and vu = d of the corresponding commutative homol- e2πiθuv for a fixed real number θ ∈ [0, 1] . ogy class (it is of the order of 3d). There is also Each pair {u, u∗} and {v,v∗} of generators a degree filtration on the K-homology for lo- generates a commutative subalgebra isomor- cally convex algebras. The K-homology degree ∞ phic to the algebra of C -functions on the of a K-homology class for CΣ is the same as circle. If θ =0, then these subalgebras com- the dimension of the corresponding commu- mute and Aθ will be isomorphic to the alge- tative homology class. bra of C∞-functions on the 2-torus S1 × S1. The K-theory for Aθ contains two even- and two odd-dimensional classes. The cyclic theory shows more precisely that one of the even classes is 0-dimensional and the other one is 2-dimensional. The two odd classes are both 1-dimensional (and are represented by the 1-forms u−1du and v−1dv). Thus, from this point of view the noncommutative torus looks exactly like an ordinary 2-torus.

In these examples the K-theory and the cyclic classes have been used to describe the “shape” of a noncommutative space. This is by no means the only function of these invariants. They also are the main tool to describe other topological information, such as gluing data in extensions and indices of operators. In this article we sketch a uniform approach to cyclic theory and bivariant K-theory, which in fact 4. Noncommutative simplicial complexes. Let Σ can be made to work for many different categories be a finite simplicial complex given by its set of algebras. This approach emphasizes the analogy of vertices V and by its simplices, represented of cyclic theory with de Rham theory and the con- by finite subsets F of V. We can associate with nection between K-theory and extensions. It leads C Σ a noncommutative algebra Σ in the fol- in a natural way to the fundamental properties of C lowing way. Let Σ be the unital algebra given both theories. We will also explain the construction by power series with rapidly decreasing coef- of the bivariant Chern-Connes character taking ∈ ficients in generators hs (s V) satisfying the bivariant K-theory to bivariant cyclic theory. The following relations: existence of this multiplicative transformation has been obtained in full generality only very recently • ∈ hs =1; s V (important special cases had been considered by Connes and others, e.g., [4], [12]) as a result of • if {s0,s1,...,sn} is not in Σ, then the progress both on the cyclic homology side and on product hs hs ...hs is zero. 0 1 n the K-theory side [9], [6]. It is a vast generalization (Note that when we introduce the additional of the classical Chern character in differential relation that the generators commute, we get geometry and allows one to associate “character- an algebra isomorphic to an algebra of C∞ istic classes” with K-theoretic objects. functions on the geometric realization of Σ.) I am indebted to my sons, Nicolas and Michael, The K-theory and the periodic cyclic homol- for the illustrations to the examples above. Since ogy for CΣ are isomorphic respectively to the these pictures have no technical meaning, they K-theory and the Z/2-graded singular coho- are only meant to provide a kind of suggestive mology of the geometric realization of Σ. The visualization of the corresponding quantum spaces.

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1 1 1 1 The Noncommutative de Rham Complex— where Ω A is the quotient Ω A/[A, Ω A] of Ω A 1 1 or How to Extract Commutative by commutators, : Ω A → Ω A is the quotient Information from a Quantum Space map, and the boundary operators are defined by Any noncommutative algebra A can be abelianized δ(x)=(dx) and β((xdy))=[x, y]. by dividing out the ideal generated by all com- It is certainly somewhat surprising that the ex- mutators. This procedure, however, destroys the tremely simple complex X(A) should play the role relevant information in nearly all interesting cases. of the de Rham complex in noncommutative geom- In fact, the abelian quotient typically is zero (this etry. In the commutative case it obviously does not is the case in examples 1, 2, and 3 above). reduce to the de Rham complex. Its analogy with A more promising approach consists in divid- the de Rham complex will now be explained. This ing only by the linear space of commutators or, also leads to a new interpretation for the classical dually, in considering traces on A. A trace is by de Rham theory. definition a linear functional f on A such that The starting point is that even though taking f (xy)=f (yx) traces is a much milder procedure than abelianizing, it still leads to trivial results for many noncommu- for all x and y in A. To describe topological infor- tative algebras. Indeed, there are natural examples mation, the strategy then is to consider homotopy of algebras A for which no nontrivial traces exist. One classes of traces. What is homotopy for traces, and is led to consider also traces on algebras related to how can this be formulated algebraically? A via an extension. An extension of A is an algebra An answer is provided by the simple X-complex E that admits A as a quotient (by an ideal I) or, in introduced by Quillen [14] in connection with short, an exact sequence differential graded algebras and then used system- atically in [7], [8], [9]. 0 −→ I −→ E −→ A −→ 0, Let A be an algebra. The space Ω1A of abstract where the arrows are algebra homomorphisms. 1-forms over A is defined as the bimodule Extensions play a fundamental role in noncom- consisting of linear combinations of expressions mutative geometry. of the form xd(y) with x ∈ Ã and y ∈ A, where Ã An algebra A is called quasi-free if in any ex- is the unitization of A. The bimodule structure is tension of A where the ideal I is nilpotent (that is, given by the rules Ik =0 for some k), there is a homomorphism a(xdy)=axdy (xdy)a = xd(ya) − xyda, a ∈ A A → E that is a left inverse for the quotient map (that is, one introduces the relation d(xy)= E → A. This condition is formally nearly identical xdy + d(x)y). A trace on Ω1A is a linear functional to the condition of smoothness introduced for f such that f (aω)=f (ωa) for a ∈ A and ω ∈ Ω1A. commutative algebras and algebraic varieties by The (dual) X-complex X (A) is the Z/2-graded Grothendieck. Any free algebra is quasi-free. ∗ complex The periodic cyclic cohomology HP (A) of an algebra A, where ∗ = ev/od, is obtained by rep- −→β {functionals on A} ←− {traces on Ω1A}. resenting A as a quotient of a quasi-free algebra δ T by an ideal I and then writing The boundary operators are defined by (δf )(x) = ∗ ∗ n f (dx) and (βf )(xdy)=f ([x, y]). It is straightforward HP (A) = lim−−→ HX (T/I ). to check that βδ = δβ =0. n This complex has only two homology groups, It is an important fact that this definition does not namely, the even and the odd homology. The even depend on the choice of the quasi-free homology HXev is the space of traces—linear resolution T. The homology HX∗(T/In) is essen- functionals on A that vanish on commutators— tially the (non-periodic) ordinary cyclic cohomol- divided by the space of “derivatives” of traces— ogy HC2n+∗(A). linear functionals of the form f ◦ d, where f is a trace It is not very difficult to see that one can alter- on Ω1A. This quotient can be considered rightfully natively obtain HP∗ by the formula as the space of homotopy classes of traces on A. ∗ ∗+1 n Arguing similarly for the odd homology, we obtain (1) HP (A) = lim−−→ HX (I ) n ev { } HX (A)= homotopy classes of traces on A where ev +1=od and vice versa. In this picture a HXod(A)={classes of closed traces on Ω1A} cyclic cocycle (of dimension 2n − 1) is described by a trace on the n-th power of the ideal in an where a trace f is closed if f ◦ d =0. extension of A. There is a natural complex X(A) for which X (A) This definition of cyclic homology is manifestly is the dual: namely, quite different from the original definition by δ −→ 1 Connes or Tsygan. The proof that both definitions A ←− Ω A β give the same result is nontrivial. Note that any

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algebra A has a canonical quasi-free (even free) Embedding Quantum Spaces into Smooth resolution given by the tensor algebra TA over A. Spaces—Extensions and K-Theory There is a striking analogy between periodic Over the years (see, e.g., [3], [11], [4], [6]) it has cyclic homology and Grothendieck’s notion of in- become evident that the single most important finitesimal homology which describes the topology notion in noncommutative topology is the one of nonsmooth varieties in algebraic geometry. In of an extension. As already mentioned above, an this analogy quasi-free algebras play the role, in extension of an algebra A is an exact sequence the noncommutative category, of smooth varieties, 0 −→ I −→ E −→ A −→ 0 and the X-complex corresponds to the de Rham complex. where the arrows are algebra morphisms. Dually, In fact, in algebraic geometry the infinitesimal and intuitively, such an extension corresponds to cohomology is defined by writing the coordinate embedding the quantum space corresponding to ring A of the variety as a quotient S/I of the co- A into the quantum space corresponding to E. ordinate ring of a smooth variety S and by taking To illustrate the kind of topological information the de Rham cohomology of the completion contained in an extension, we mention that the n content of the Atiyah-Singer index theorem [1] S =← lim−− S/I . It can be shown that this does not n may be viewed as the determination of the class depend on the choice of the embedding into a of the extension defined by the pseudodifferential smooth variety. This procedure is exactly analogous operators on a compact manifold. Other index to the definition of periodic cyclic cohomology theorems are concerned with more complicated above, where we write A as a quotient of a quasi- extensions. free algebra T and then take the homology HX∗(T). We are now going to sketch a general con- Given two algebras A1 and A2, we can also de- struction, based on extensions, of (bivariant) fine the bivariant periodic cyclic theory HP∗(A1,A2), topological K-theory that works for many cate- where ∗ =0, 1 , as the homology of the Hom- gories of noncommutative algebras and fits complex between the X-complexes associated with nicely with the definition of cyclic theory outlined quasi-free extensions for A1 and A2 . There is a above. As a result there will be a natural transfor- natural composition product mation, the bivariant Chern-Connes character, from topological K-theory to cyclic theory. HPi(A1,A2) × HPj (A2,A3) −→ HPi+j (A1,A3). To be more specific, we will assume from now on that all our algebras are topological algebras Any algebra homomorphism α : A → B determines with a complete locally convex structure given an element HP(α) in HP (A, B). 0 by a family of seminorms (p ) satisfying The periodic cyclic theory HP (A ,A ) has very α i 1 2 p (xy) ≤ p (x)p (y) for all x and y in the algebra. good properties. It can be shown that it is invari- α α α For any such locally convex algebra A, the algebraic ant under differentiable homotopies and under tensor algebra A ⊕ A⊗2 ⊕ A⊗3 ⊕ ... has a natural Morita invariance in both variables. Moreover, by completion TA which is a locally convex algebra. [9] it satisfies “excision” in the following sense. There is a canonical algebra homomorphism Let D be any algebra. Every extension TA → A mapping x1 ⊗ ...⊗ xn to the product i q 0 → I → A → B → 0 induces exact sequences in x1x2 ...xn. We denote the kernel by JA, so that we HP∗(D, · ) and HP∗( · ,D) as follows: get a free resolution for A of the form 0 −→ JA −→ TA −→ A −→ 0, ·HP(i) ·HP(q) HP0(D, I) −→ HP0(D, A) −→ HP0(D, B) which can be viewed as an embedding of the quan- ↑↓tum space corresponding to A into the smooth quantum space corresponding to TA. The ideal ·HP(q) ·HP(i) HP1(D, B) ←− HP1(D, A) ←− HP1(D, I) corresponds to the complement of the image of the quantum space for A. Since JA is again a locally and convex algebra, we can iterate the construction and form J2A = J(JA) and, inductively, JnA = n−1 HP(q)· HP(i)· J(J A). HP (I,D) ←− HP (A, D) ←− HP (B,D) 0 0 0 With a locally convex algebra B, we can also ↓↑associate the algebra M∞(B) of infinite matrices · HP(i)· HP(q) (bij)i,j∈N with rapidly decreasing matrix elements HP1(B,D) −→ HP1(A, D) −→ HP1(I,D) in B. The corresponding quantum space looks like the one in example (1) of the introduction. It has Special cases of these exact sequences had been infinitely many points, which are labeled by N, obtained before in [10] and [16]. They are one of and arrows between the points, which are labeled the main tools in the computation of the cyclic by all possible elements of B. A fundamental homology invariants. extension, using pseudodifferential operators on

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the circle, shows that for any A there exists a The pseudodifferential operators on a smooth canonical homomorphism J2A → M∞(B). This map compact manifold give rise to an extension can be used to form the inductive limit in the ∞ ∗ 0 −→ Ψ− −→ Ψ −→ C (S M) −→ 0, following definition. 1 0 Let A and B be locally convex algebras, and where Ψ−1 and Ψ0 denote the algebras of pseudo- ∗ =0 or 1. We define differential operators of order −1 and 0, respec- ∞ ∗ 2k+∗ tively, and C (S M) denotes the algebra of smooth kk∗(A, B) = lim−−→[J A, M∞(B)], k functions on the cosphere-bundle of M . The ∗ problem solved by the Atiyah-Singer index where [J2k+ A, M∞(B)] denotes the set of (differ- theorem is exactly the determination of the class entiable) homotopy classes of homomorphisms ∞ ∗ ∗ in kk1(C (S M), Ψ−1) defined by this extension. J2k+ A → M∞(B). Provided with the ordinary direct sum addition of maps into matrices, kk∗(A, B) is The Bivariant Chern-Connes Character an abelian group. The most important ingredient in the construction X Since taking the homology of the -complex is of a bivariant multiplicative transformation from an algebraic version of taking homotopy classes, kk∗ to the bivariant theory HP∗ on the category this definition is formally remarkably similar to the of locally convex algebras is the universality prop- definition of periodic cyclic theory in equation (1) erty of kk0 mentioned at the end of the preceding in the preceding section. section. This bifunctor kk∗ has the same abstract ∗ Since HP0 satisfies the properties for which kk0 properties as HP (see the preceding section). In is universal, we immediately obtain a transforma- particular: tion • Every homomorphism α : A → B induces an ch : kk0(A, B) −→ HP0(A, B) element kk(α) in kk0(A, B). • There is an associative product kki(A, B) which is compatible with the product. × kkj (B,C) → kki+j (A, C) (where i,j ∈ Z/2 , In trying to extend this to a multiplicative trans- and A, B, and C are locally convex algebras) formation from the Z/2-graded theory kk∗ to that is additive in both variables and that HP∗, one faces the problem that the product of two satisfies kk(α)kk(β)=kk(αβ) for two homo- odd classes is defined differently in kk∗ and in morphisms α and β. HP∗ . It turns out that one has to√ introduce (some- • kk∗ is homotopy invariant and satisfies what arbitrarily) a factor of 2πi. With this excision in both variables (as described in the proviso one does then obtain a transformation preceding section). The canonical map ch : kk∗(A, B) −→ HP∗(A, B) B → M∞(B) induces an isomorphism in both variables of kk∗. which is multiplicative in full generality. ∗ In fact, one can show that kk0 is the universal Both the cyclic cohomology HP (A) and the K- functor from the category of locally convex alge- homology kk∗(A, C) have a natural (dimension) bras as above into an additive category satisfying filtration due to their definitions as inductive the third property. limits. This dimension filtration was alluded to in Even though this construction of K-theory looks the examples in the introduction. really different from the usual approach using The behaviour of these filtrations under the projections or projective modules, it turns out Chern-Connes character is well understood due that when we specialize the first variable to a to a very delicate analysis of the boundary map point, i.e., to the algebra C of complex numbers, in the cyclic homology long exact sequence by kk∗(C,B) is nothing but the usual K-group in the M. Puschnigg and by R. Meyer. Given an element case where B is a Banach algebra [2] or a Fréchet α in kk∗(A, C), the dimension of ch(α) is bounded algebra [13] (the cases in which the usual K- by 3d, where d is the K-theoretic dimension of α. theory is defined). This estimate is optimal. Another important property is that any exten- To close this article, we want to emphasize that sion, or more generally any “n-step” extension of despite their seemingly abstract definition, the the form cyclic theory and K-theory invariants can be computed very explicitly for a large variety of 0 → B → E →···→E → A → 0, 1 n noncommutative algebras. Some typical examples gives an element in kkn(A, B) where n is counted were described in the introduction. modulo 2. In homological algebra one uses a well-known product on such extensions, known as References the Yoneda product, which consists simply [1] M. F. ATIYAH and I. M. SINGER, The index of elliptic op- in splicing together two such extensions. This erators I, Ann. of Math. (2) 87 (1968), 484–530. product is compatible with the product [2] B. BLACKADAR, K-theory for Operator Algebras, Springer-Verlag, Heidelberg-Berlin-New York-Tokyo, kkn(A, B) × kkm(B,C) → kkn+m(A, C). 1986.

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[3] L. G. BROWN, R. G. DOUGLAS, and P. FILLMORE, Extensions of C*-algebras and K-homology, Ann. of Math. 105 About the Cover (1977), 265–324. The image on this month's cover arose from [4] A. CONNES, Non-commutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1985), Joachim Cuntz's effort to render into visible art 257–360. his own internal vision of a noncommutative torus, an object otherwise quite abstract. His [5] ——— , Non-commutative Geometry, Academic Press, London-Sydney-Tokyo-Toronto, 1994. original idea was then implemented by his son [6] J. CUNTZ, Bivariante K-theorie für lokalkonvexe Michael in a program written in Pascal. More Algebren und der bivariante Chern-Connes- explicitly, he says that the construction started Charakter, Doc. Math. J. DMV 2 (1997), 139–182; out with a triangle in a square, then translated http://www.mathematik.uni-bielefeld.de/ the triangle by integers times a unit along a line documenta/. with irrational slope; plotted the images thus [7] J. CUNTZ and D. QUILLEN, Algebra extensions and non- obtained in a periodic manner; and stopped singularity, J. Amer. Math. Soc. 8 (1995), 251–289. just before the figure started to seem clut- [8] ——— , Cyclic homology and nonsingularity, J. Amer. Math. Soc. 8 (1995), 373–442. tered. Many mathematicians carry around inside [9] ——— , Excision in bivariant periodic cyclic cohomology, Invent. Math. 127 (1997), 67–98. their heads mental images of the abstractions [10] T. G. GOODWILLIE, Cyclic homology and the free loop- they work with, and manipulate these objects space, Topology 24 (1985), 187–215. somehow in conformity with their mental im- [11] G. G. KASPAROV, The operator K-functor and exten- agery. They probably also make aesthetic judge- sions of C*-algebras (in Russian), Izv. Akad. Nauk. ments of the value of their work according to SSSR Ser. Mat. 44 (1980), 571–636; Math. USSR Izv. the visual qualities of the images. These pre- 16 (1981), 513-572. sumably common phenomena remain a rarely [12] V. NISTOR, A bivariant Chern-Connes character, Ann. of Math. 138 (1993), 555–590. explored domain in either art or psychology. [13] C. PHILLIPS, K-theory for Fréchet algebras, Internat. J. —Bill Casselman ([email protected]) Math. 2 (1991), 77–129. [14] D. QUILLEN, Chern-Simons forms and cyclic cohomology, The Interface of Mathematics and Particle Physics, Clarendon Press, 1990, pp. 117–134. [15] B. TSYGAN, The homology of matrix Lie algebras over rings and the Hochschild homology (in Russian), Uspekhi Mat. Nauk 38 (1983), 217–218; Russian Math. Surveys 38 (1983), 198–199. [16] M. WODZICKI, Excision in cyclic homology and in rational algebraic K-theory, Ann. of Math. 129 (1989), 591–639.

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