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Non-commutative differential geometry. (English) Zbl 0592.46056

In this article, Connes lays the groundwork for a theory of noncommutative differential geometry, i.e. differential geometry for noncommutative algebras generalizing the commutative algebra 𝒞 (M) of smooth functions on a compact manifold. The idea of doing topology of noncommutative ”topological spaces”, i.e. C * - algebras, is not new and has, in fact, found a very satisfactory realization via K–theory and notably Kasparov’s KK–theory for C * - algebras. To do differential geometry, one would like to have an analogue of de Rham (co)homology and characteristic classes. This generalization, the ”cyclic cohomology” is obtained by Connes in the following way.

Let 𝔄 be a complex algebra. The differential envelope Ω 𝔄 consists of all linear combinations of abstract (not necessarily antisymmetric) ”differential forms” x 0 dx 1 ···dx n over 𝔄 (i.e. x i 𝔄) where d(xy)=xdy+d(x)y, x,y𝔄. In other words Ω 𝔄 is the universal algebra containing 𝔄 and admitting a linear map d:Ω𝔄Ω𝔄 such that d(xy)=xdy+dxy, (x,y𝔄) and d 2 =0. Let Ω n 𝔄 be the space of all such forms of degree n. Every trace T:Ω n 𝔄 (T is a trace if T(ωω ' )=T(ω ' ω) for ωΩ k 𝔄, ω ' Ω 𝔄, k+=n) which is in addition closed, i.e. T(dω)=0, ω, gives rise to an (n+1)-linear functional f(x 0 ,···,x n )=T(x 0 dx 1 ···dx n ) which has the following properties

(1) f(x 1 ,···,x n ,x 0 )=(-1) n f(x 0 ,···,x n )

(2) bf=0 where b is the Hochschild boundary operator

bf(x 0 ,···,x n+1 )=f(x 0 x 1 ,···,x n+1 )-f(x 0 ,x 1 x 2 ,···,x n+1 )+···+(-1) n+1 f(x n+1 x 0 ,···,x n )·

Let C λ n (𝔄) be the space of all f satisfying (1). Then bC λ n C λ n+1 so that one obtains a subcomplex (C λ n ,b) of the Hochschild complex on (C n (𝔄,𝔄 * ),b). The cohomology of this complex is the ”cyclic cohomology” of 𝔄- denoted by H λ n (𝔄)·

The introduction of this concept is motivated by Ext-theory and the Chern-character in K–homology. The construction of the Chern character in turn has its roots in the work of Helton and Howe and Kasparov’s KK–theory. An element of Kasparov’s K–homology group K 0 (𝔄) is described by a pair (ϕ,ϕ ¯) of *-homomorphisms of 𝔄 into (H), H a Hilbert space, such that ϕ (x)-ϕ ¯(x) is a compact operator for all x𝔄. If now (ϕ,ϕ ¯) is even such that q(x)=ϕ(x)-ϕ ¯(x) is in the Schatten class 𝒞 p+1 (H) for all x𝔄, one can define f(x 0 ,···,x p )=Tr(q(x 0 )···q(x p )) where Tr is the usual trace 𝒞 1 (H). One easily checks that fC λ p (𝔄) if p is even and one can define the Chern character ch by ch p ((ϕ,ϕ ¯))=[f]H λ p (𝔄). This construction gives rise naturally to an operator S:H λ p H λ p+2 which has the property that ch p+2 ((ϕ,ϕ ¯)) (which is also defined since 𝒞 p+1 𝒞 p+3 ) equals S ch p ((ϕ,ϕ ¯))·

One can now define H even (𝔄)=lim (H λ 0 (𝔄) S H λ 2 (𝔄) S ···) and H p (𝔄) the image of H λ p (𝔄) in H even divided by the image of H λ p-2 (𝔄) (of course, the odd case is treated similarly). Connes shows that for 𝔄=𝒞 (M), M a smooth compact manifold, one finds that H n (𝔄) is equal to the de Rham homology group H n (M,). He also establishes a long exact sequence

···H n+1 (𝔄,𝔄 * )H λ n (𝔄) S H λ n+2 (𝔄)H n+2 (𝔄,𝔄 * )···

connecting cyclic cohomology with Hochschild cohomology, and uses this sequence for instance to compute H * (𝔄) for the canonical dense subalgebra of the ”irrational rotation algebra”.

The article contains, in addition, a wealth of information which is impossible to describe in a brief review. With cyclic cohomology we dispose of a completely new, unexpected and powerful tool opening many new roads in non-commutative topology, homological algebra, algebraic K–theory and probably also classical differential geometry.

Reviewer: J. Cuntz

MSC:
46L87Noncommutative differential geometry
46L89Other “noncommutative” mathematics based on C * -algebra theory
46L85Noncommutative topology
58B32Geometry of quantum groups
58B34Noncommutative geometry (á la Connes)
18G35Chain complexes (homological algebra)
18G60Other (co)homology theories
58A12de Rham theory (global analysis)
18F25Algebraic K-theory and L-theory
19D55K-theory and homology; cyclic homology and cohomology
19K35Kasparov theory (KK-theory)
19K56Index theory (K-theory)
58J20Index theory and related fixed point theorems (PDE on manifolds)
58J22Exotic index theories (PDE on manifolds)
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