fea-cuntz.qxp 8/2/01 9:03 AM Page 793 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. SEPTEMBER 2001 NOTICES OF THE AMS 793 fea-cuntz.qxp 8/2/01 9:03 AM Page 794 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 infor- mation 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 noncommuta- tive connections. This space has n points and arrows between every two points. As an alge- bra 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) gen- erated 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 the- ory 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 the- ories 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 con- nected 0-dimensional space. This is the simplest case of a convolution algebra for a groupoid. In general a (topolog- ical) groupoid consists of a space of objects and a family of (invertible) arrows which can be con- sidered as above as noncommutative paths between the objects. For the noncommutative homology theories, different points connected by an arrow will be homologous. Higher ho- mology classes can also arise from configura- tions 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 794 NOTICES OF THE AMS VOLUME 48, NUMBER 8 fea-cuntz.qxp 8/2/01 9:03 AM Page 795 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 the- ory 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 informa- tion, 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.