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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 ${𝒞}^{\infty }\left(M\right)$ 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 ${\Omega }$ $𝔄$ consists of all linear combinations of abstract (not necessarily antisymmetric) ”differential forms” ${x}_{0}d{x}_{1}···d{x}_{n}$ over $𝔄$ (i.e. ${x}_{i}\in 𝔄\right)$ where $d\left(xy\right)=xdy+d\left(x\right)y$, x,y$\in 𝔄$. In other words ${\Omega }$ $𝔄$ is the universal algebra containing $𝔄$ and admitting a linear map $d:{\Omega }𝔄\to {\Omega }𝔄$ such that $d\left(xy\right)=xdy+dxy$, $\left(x,y\in 𝔄\right)$ and ${d}^{2}=0$. Let ${{\Omega }}^{n}𝔄$ be the space of all such forms of degree $n$. Every trace $T:{{\Omega }}^{n}𝔄\to ℂ$ ($T$ is a trace if $T\left(\omega {\omega }^{\text{'}}\right)=T\left({\omega }^{\text{'}}\omega \right)$ for $\omega \in {{\Omega }}^{k}𝔄$, ${\omega }^{\text{'}}\in {{\Omega }}^{\ell }𝔄$, $k+\ell =n\right)$ which is in addition closed, i.e. $T\left(d\omega \right)=0$, $\forall \omega$, gives rise to an $\left(n+1\right)$-linear functional $f\left({x}_{0},···,{x}_{n}\right)=T\left({x}_{0}d{x}_{1}···d{x}_{n}\right)$ which has the following properties

(1) $f\left({x}_{1},···,{x}_{n},{x}_{0}\right)={\left(-1\right)}^{n}f\left({x}_{0},···,{x}_{n}\right)$

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

$bf\left({x}_{0},···,{x}_{n+1}\right)=f\left({x}_{0}{x}_{1},···,{x}_{n+1}\right)-f\left({x}_{0},{x}_{1}{x}_{2},···,{x}_{n+1}\right)+···+{\left(-1\right)}^{n+1}f\left({x}_{n+1}{x}_{0},···,{x}_{n}\right)·$

Let ${C}_{\lambda }^{n}\left(𝔄\right)$ be the space of all $f$ satisfying (1). Then $b{C}_{\lambda }^{n}\subset {C}_{\lambda }^{n+1}$ so that one obtains a subcomplex $\left({C}_{\lambda }^{n},b\right)$ of the Hochschild complex on $\left({C}^{n}\left(𝔄,{𝔄}^{*}\right),b\right)$. The cohomology of this complex is the ”cyclic cohomology” of $𝔄$- denoted by ${H}_{\lambda }^{n}\left(𝔄\right)·$

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 ${K}^{0}\left(𝔄\right)$ is described by a pair ($\phi$,$\overline{\phi }$) of *-homomorphisms of $𝔄$ into $ℬ\left(H\right)$, $H$ a Hilbert space, such that $\phi$ (x)-$\overline{\phi }$(x) is a compact operator for all $x\in 𝔄$. If now ($\phi$,$\overline{\phi }$) is even such that $q\left(x\right)=\phi \left(x\right)-\overline{\phi }\left(x\right)$ is in the Schatten class ${𝒞}^{p+1}\left(H\right)$ for all $x\in 𝔄$, one can define $f\left({x}_{0},···,{x}_{p}\right)=\text{Tr}\left(q\left({x}_{0}\right)···q\left({x}_{p}\right)\right)$ where $\text{Tr}$ is the usual trace ${𝒞}^{1}\left(H\right)\to ℂ$. One easily checks that $f\in {C}_{\lambda }^{p}\left(𝔄\right)$ if p is even and one can define the Chern character ch by $c{h}_{p}\left(\left(\phi ,\overline{\phi }\right)\right)=\left[f\right]\in {H}_{\lambda }^{p}\left(𝔄\right)$. This construction gives rise naturally to an operator $S:{H}_{\lambda }^{p}\to {H}_{\lambda }^{p+2}$ which has the property that $c{h}_{p+2}\left(\left(\phi ,\overline{\phi }\right)\right)$ (which is also defined since ${𝒞}^{p+1}\subset {𝒞}^{p+3}\right)$ equals S ch${}_{p}\left(\left(\phi ,\overline{\phi }\right)\right)·$

One can now define ${H}^{even}\left(𝔄\right)={lim}_{\to }\left({H}_{\lambda }^{0}\left(𝔄\right){\to }^{S}{H}_{\lambda \phantom{\rule{1.em}{0ex}}}^{2}\left(𝔄\right){\to }^{S}···\right)$ and ${H}^{p}\left(𝔄\right)$ the image of ${H}_{\lambda }^{p}\left(𝔄\right)$ in ${H}^{even}$ divided by the image of ${H}_{\lambda }^{p-2}\left(𝔄\right)$ (of course, the odd case is treated similarly). Connes shows that for $𝔄={𝒞}^{\infty }\left(M\right)$, $M$ a smooth compact manifold, one finds that ${H}^{n}\left(𝔄\right)$ is equal to the de Rham homology group ${H}_{n}\left(M,ℂ\right)$. He also establishes a long exact sequence

$···\to {H}^{n+1}\left(𝔄,{𝔄}^{*}\right)\to {H}_{\lambda }^{n}\left(𝔄\right){\to }^{S}{H}_{\lambda }^{n+2}\left(𝔄\right)\to {H}^{n+2}\left(𝔄,{𝔄}^{*}\right)\to ···$

connecting cyclic cohomology with Hochschild cohomology, and uses this sequence for instance to compute ${H}^{*}\left(𝔄\right)$ 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 (á la Connes) 18G35 Chain complexes (homological algebra) 18G60 Other (co)homology theories 58A12 de Rham theory (global analysis) 18F25 Algebraic $K$-theory and $L$-theory 19D55 $K$-theory and homology; cyclic homology and cohomology 19K35 Kasparov theory ($KK$-theory) 19K56 Index theory ($K$-theory) 58J20 Index theory and related fixed point theorems (PDE on manifolds) 58J22 Exotic index theories (PDE on manifolds)