jones.qxp 3/25/98 9:00 AM Page 792 Review of Noncommutative Geometry by Alain Connes Vaughan Jones and Henri Moscovici This and the following article are expositions C(X) of all continuous complex-valued func- of Noncommutative Geometry by Alain Connes tions on X (which, by the famous Gelfand- (Academic Press, 1994, 661 pages, $59.95 hard- Naimark theorem, completely captures the topol- cover), reviewed in the Bulletin of the Ameri- ogy of a compact Hausdorff space). Then replace can Mathematical Society, v. 33 no. 4, October the quotient space not by a smaller algebra such 1996, pp. 459–466. as C(X/ ) but by a larger one, the crossed prod- uct C(X) o consisting of sums γ fγ γ (with ∈ fγ C(XΓ)) which are multiplied according to “The correspondence between geometric spaces ∈ P the rule γfγΓ 1 = f γ 1. This algebraΓ is non- and commutative algebras is a familiar and basic − ◦ − idea of algebraic geometry. The purpose of this commutative and should be completed in some book is to extend the correspondence to the non- fashion, but when this is done, it provides a commutative case in the framework of real analy- powerful tool for the study of X/ . In the con- sis.” text of operator algebras there is nothing patho- Thus begins Connes’ book Noncommutative logical about C(X) o which can beΓ thought of Geometry. The central thesis is that the usual no- as a noncommutative desingularization of X/ . tion of a “space”—a set with some extra struc- In our example theΓ noncommutative algebra ture—is inadequate in many interesting cases was obtained from a group acting on a com-Γ and that coordinates may profitably be replaced mutative one, but once one admits the legitimacy by a noncommutative algebra. Here is a simple of such algebras as “coordinate rings”, one may but central example. Suppose the compact Haus- expect to find new examples devoid of any com- dorff space X is acted on by a discrete group . mutative origin. If the action is sufficiently simple, one may put Connes explores many examples, as diverse the quotient topology on the orbit space, X/Γ, as the space of Penrose tilings and the Brillouin thus capturing the orbit structure in an entirely zone in the quantum Hall effect. In so doing he satisfactory fashion. But as the action becomesΓ develops a myriad of techniques, sometimes, as more complicated, the quotient topology will in the case of his Chern character, as necessary fail to separate orbits, and in extreme cases like generalizations of known tools and sometimes, the action of Z by powers of an irrational rota- as in the case of his quantized calculus, of largely tion of the circle the quotient topology has no noncommutative inspiration. information whatsoever. The noncommutative The technical background of Connes’ pro- approach is first to replace X by the algebra gram is the theory of operator algebras on Hilbert space, closed in some topology. Why should op- Vaughan Jones is professor of mathematics at the Uni- erator algebras be the vessel of noncommutative versity of California, Berkeley. Henri Moscovici is pro- geometry? After all, the Gelfand-Naimark theo- fessor of mathematics at The Ohio State University. rem is not about operators, and, according to 792 NOTICES OF THE AMS VOLUME 44, NUMBER 7 jones.qxp 3/25/98 9:00 AM Page 793 what we have said so far, it is little more than a mann algebra is X⊕ M(ξ)dµ(ξ) for some mea- curiosity that may faithfully represent C(X) on sure space (X,µ) withR each M(ξ) a factor. Fac- a Hilbert space. There is a simple but telling rea- tors themselves were classified in [22] accord- son for operator algebras. The first noncom- ing to algebraic properties of the projections mutative algebras are the n n matrix algebras (operators p with p2 = p = p ) which they con- × ∗ Mn(C), which are algebras of operators on finite- tain. In a von Neumann algebra M we say p q dimensional Hilbert space. So operator algebra if pq = p and p q if there is a u M with≤ ∼ ∈ is just “large matrix algebra”. Or we could take uu∗ = p and u∗u = q. There is a direct analogy a cue from quantum physics, where the classi- with cardinalities of sets, so a projection is called cal observables—continuous functions on phase “infinite” if it is equivalent to a proper subpro- space—are replaced by algebras of operators, in jection, and “ ” induces an order “.” on equiv- general unbounded, on the Hilbert space of wave alence classes≤ of projections in M. If M were the functions. According to von Neumann’s re- n n matrices, we would have p q if and only × ∼ markable insight, algebras of bounded opera- if rank(p)=rank(q). In a factor, . is a total tors, if sufficiently well understood, would in fact order. The factor’s type is determined by the ex- be enough to understand the unbounded ones. istence of infinite projections type II and III, ∞ This led to “algebraic quantum field theory” ini- and the type of the ordered set . according to tiated by Haag and Kastler in [14]. But we digress. the following table (the Hilbert space is assumed In algebraic geometry access to the power of separable). the theory requires nontrivial results in pure commutative algebra. In noncommutative geom- etry, then, we expect to need a serious dose of operator algebraic results right at the start. Far more so indeed than algebraic geometry needs commutative algebra, since the intuition af- forded by the set of points in a variety has dis- appeared and there is no reason to expect that all the interesting phenomena of noncommuta- Here is an enormously condensed summary tive geometry arise as mere perturbations of of the work of Murray and von Neumann. commutative ones. Thus a substantial, but by no • Any type I factor is of the form ( 1) id B H ⊗ means dominant, portion of Connes’ book is de- in some tensor “factorization” voted to the abstract structure theory of opera- = 1 2 (where we use ( ) to de- H H ⊗H B · tor algebras. note the algebra of all bounded operators). Operator algebra theory splits almost dis- • Type II 1 factors may be constructed as the jointly into two halves: von Neumann algebras, commutant of the regular representation of initiated by Murray and von Neumann in the a discrete group all of whose conjugacy 1930s [22], and C*-algebras, initiated by Gelfand classes are infinite. and Naimark in the 1940s [12]. Technically the • Type II factors are always of the form II 1 ∞ ⊗ split is according to the topology in which the I (i.e., infinite matrices over a II 1 factor). ∞ algebras are closed-von Neumann algebras in • Type III factors are more difficult but were the topology of pointwise convergence and C*- constructed in [30]. algebras in the norm topology. But morally the • Type II 1 factors afford a trace—a linear split is the noncommutative version of the split functional tr: M C with tr(x∗x) > 0 for → between measure theory and general topology. x =0 and tr(xy)= tr(yx). Type II factors 6 ∞ If functions on [0,1] are represented on have an infinite trace analogous to that on L2([0, 1],dx) as multiplication operators, the ( ), and type III factors have no trace B H continuous functions form a C*-algebra and the whatsoever. L∞ functions form a von Neumann algebra. • There is, up to isomorphism, a single II 1 Every von Neumann algebra is a C*-algebra, but factor with the “hyperfinite” property; i.e., it is seldom wise to think of it as such. any finite set of elements can be arbitrarily Let us first discuss von Neumann algebras, well approximated by elements in a finite- where Connes has been such a dominant figure dimensional subalgebra. And there are II 1 for the last twenty-five years. Murray and von factors without this property. Neumann quickly recognized the importance of The period from the mid 1940s to the late von Neumann algebras whose center contains 1960s was a period of digestion of the results only scalar multiples of the identity, which they of Murray and von Neumann. Highlights include called “factors”, and von Neumann wrote a paper the discovery by Dixmier of a plethora of max- [30] reducing any von Neumann algebra to a fac- imal abelian subalgebras of a II 1 factor [10], the tor by a “direct integral”—a continuous ana- work of Dye on orbit equivalence [11], estab- logue of the direct sum of algebras. Any von Neu- lishing the hyperfiniteness of crossed products AUGUST 1997 NOTICES OF THE AMS 793 jones.qxp 3/25/98 9:00 AM Page 794 of abelian algebras by Z, and the characteriza- the kernel of the map from R to Out(M). Now tion by Sakai of von Neumann algebras as C*-al- T (M) can be quite complicated, and Connes also gebras which are duals as Banach spaces [26]. Let defined an invariant S(M) as me cite two results as being the ultimate re- ϕ spectrum( ϕ) 0 , which is a closed multi- \{ } + finement of the Murray-von Neumann ideas—the plicativeT subgroup of R . A type III factor is said construction by McDuff of uncountably many II 1 to be of type∆ III , 0 λ 1 , according to λ ≤ ≤ factors [21] and the construction by Powers of whether S(M) is a continuum of type III factors [25].