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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 $${\mathcal C}^{\infty}(M)$$ 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 $${\mathfrak A}$$ be a complex algebra. The differential envelope $$\Omega$$ $${\mathfrak A}$$ consists of all linear combinations of abstract (not necessarily antisymmetric) ”differential forms” $$x_ 0dx_ 1...dx_ n$$ over $${\mathfrak A}$$ (i.e. $$x_ i\in {\mathfrak A})$$ where $$d(xy)=xdy+d(x)y$$, x,y$$\in {\mathfrak A}$$. In other words $$\Omega$$ $${\mathfrak A}$$ is the universal algebra containing $${\mathfrak A}$$ and admitting a linear map $$d: \Omega{\mathfrak A}\to \Omega{\mathfrak A}$$ such that $$d(xy)=xdy+dx y$$, $$(x,y\in {\mathfrak A})$$ and $$d^ 2=0$$. Let $$\Omega^ n{\mathfrak A}$$ be the space of all such forms of degree $$n$$. Every trace $$T: \Omega^ n{\mathfrak A}\to {\mathbb C}$$ ($$T$$ is a trace if $$T(\omega\omega')= T(\omega'\omega)$$ for $$\omega \in \Omega^ k{\mathfrak A}$$, $$\omega'\in \Omega^{\ell}{\mathfrak A}$$, $$k+\ell =n)$$ which is in addition closed, i.e. $$T(d\omega)=0$$, $$\forall \omega$$, gives rise to an $$(n+1)$$-linear functional $$f(x_ 0,...,x_ n)=T(x_ 0dx_ 1...dx_ n)$$ which has the following properties
(1) $$f(x_ 1,...,x_ n,x_ 0)=(-1)^ nf(x_ 0,...,x_ n)$$
(2) $$bf=0$$ where $$b$$ is the Hochschild boundary operator
$$bf(x_ 0,...,x_{n+1})=f(x_ 0x_ 1,...,x_{n+1})-f(x_ 0,x_ 1x_ 2, ...,x_{n+1})+...+(-1)^{n+1}f(x_{n+1}x_ 0,...,x_ n).$$
Let $$C^ n_{\lambda}({\mathfrak A})$$ be the space of all $$f$$ satisfying (1). Then $$bC^ n_{\lambda}\subset C_{\lambda}^{n+1}$$ so that one obtains a subcomplex $$(C^ n_{\lambda},b)$$ of the Hochschild complex on $$(C^ n({\mathfrak A},{\mathfrak A}^*),b)$$. The cohomology of this complex is the ”cyclic cohomology” of $${\mathfrak A}$$- denoted by $$H^ n_{\lambda}({\mathfrak A}).$$
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 $$K^ 0({\mathfrak A})$$ is described by a pair ($$\phi$$,$${\bar \phi}$$) of *-homomorphisms of $${\mathfrak A}$$ into $${\mathcal B}(H)$$, $$H$$ a Hilbert space, such that $$\phi$$ (x)-$${\bar \phi}$$(x) is a compact operator for all $$x\in {\mathfrak A}$$. If now ($$\phi$$,$${\bar \phi}$$) is even such that $$q(x)=\phi (x)-{\bar \phi}(x)$$ is in the Schatten class $${\mathcal C}^{p+1}(H)$$ for all $$x\in {\mathfrak A}$$, one can define $$f(x_ 0,...,x_ p)=\text{Tr}(q(x_ 0)...q(x_ p))$$ where $$\text{Tr}$$ is the usual trace $${\mathcal C}^ 1(H)\to {\mathbb{C}}$$. One easily checks that $$f\in C^ p_{\lambda}({\mathfrak A})$$ if p is even and one can define the Chern character ch by $$ch_ p((\phi,{\bar \phi}))=[f]\in H^ p_{\lambda}({\mathfrak A})$$. This construction gives rise naturally to an operator $$S: H^ p_{\lambda}\to H_{\lambda}^{p+2}$$ which has the property that $$ch_{p+2}((\phi,{\bar \phi}))$$ (which is also defined since $${\mathcal C}^{p+1}\subset {\mathcal C}^{p+3})$$ equals S ch$${}_ p((\phi,{\bar \phi})).$$
One can now define $$H^{even}({\mathfrak A})=\lim_{\to}(H^ 0_{\lambda}({\mathfrak A})\to^{S}H^ 2_{\lambda \quad}({\mathfrak A})\to^{S}...)$$ and $$H^ p({\mathfrak A})$$ the image of $$H^ p_{\lambda}({\mathfrak A})$$ in $$H^{even}$$ divided by the image of $$H_{\lambda}^{p-2}({\mathfrak A})$$ (of course, the odd case is treated similarly). Connes shows that for $${\mathfrak A}={\mathcal C}^{\infty}(M)$$, $$M$$ a smooth compact manifold, one finds that $$H^ n({\mathfrak A})$$ is equal to the de Rham homology group $$H_ n(M,{\mathbb C})$$. He also establishes a long exact sequence $...\to H^{n+1}({\mathfrak A},{\mathfrak A}^*)\to H^ n_{\lambda}({\mathfrak A})\to^{S}H_{\lambda}^{n+2}({\mathfrak A})\to H^{n+2}({\mathfrak A},{\mathfrak A}^*)\to...$ connecting cyclic cohomology with Hochschild cohomology, and uses this sequence for instance to compute $$H^*({\mathfrak A})$$ 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:
 46L87 Noncommutative differential geometry 46L89 Other “noncommutative” mathematics based on $$C^*$$-algebra theory 46L85 Noncommutative topology 58B32 Geometry of quantum groups 58B34 Noncommutative geometry (à la Connes) 18G35 Chain complexes (category-theoretic aspects), dg categories 18G60 Other (co)homology theories (MSC2010) 58A12 de Rham theory in global analysis 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 19D55 $$K$$-theory and homology; cyclic homology and cohomology 19K35 Kasparov theory ($$KK$$-theory) 19K56 Index theory 58J20 Index theory and related fixed-point theorems on manifolds 58J22 Exotic index theories on manifolds
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