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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 ${\cal C}\sp{\infty}(M)$ of smooth functions on a compact manifold. The idea of doing topology of noncommutative ”topological spaces”, i.e. $C\sp*$- algebras, is not new and has, in fact, found a very satisfactory realization via $K$--theory and notably Kasparov’s $KK$--theory for $C\sp*$- 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 ${\frak A}$ be a complex algebra. The differential envelope $\Omega$ ${\frak A}$ consists of all linear combinations of abstract (not necessarily antisymmetric) ”differential forms” $x\sb 0dx\sb 1...dx\sb n$ over ${\frak A}$ (i.e. $x\sb i\in {\frak A})$ where $d(xy)=xdy+d(x)y$, x,y$\in {\frak A}$. In other words $\Omega$ ${\frak A}$ is the universal algebra containing ${\frak A}$ and admitting a linear map $d: \Omega{\frak A}\to \Omega{\frak A}$ such that $d(xy)=xdy+dx y$, $(x,y\in {\frak A})$ and $d\sp 2=0$. Let $\Omega\sp n{\frak A}$ be the space of all such forms of degree $n$. Every trace $T: \Omega\sp n{\frak A}\to {\Bbb C}$ ($T$ is a trace if $T(\omega\omega')= T(\omega'\omega)$ for $\omega \in \Omega\sp k{\frak A}$, $\omega'\in \Omega\sp{\ell}{\frak A}$, $k+\ell =n)$ which is in addition closed, i.e. $T(d\omega)=0$, $\forall \omega$, gives rise to an $(n+1)$-linear functional $f(x\sb 0,...,x\sb n)=T(x\sb 0dx\sb 1...dx\sb n)$ which has the following properties (1) $f(x\sb 1,...,x\sb n,x\sb 0)=(-1)\sp nf(x\sb 0,...,x\sb n)$ (2) $bf=0$ where $b$ is the Hochschild boundary operator $bf(x\sb 0,...,x\sb{n+1})=f(x\sb 0x\sb 1,...,x\sb{n+1})-f(x\sb 0,x\sb 1x\sb 2, ...,x\sb{n+1})+...+(-1)\sp{n+1}f(x\sb{n+1}x\sb 0,...,x\sb n).$ Let $C\sp n\sb{\lambda}({\frak A})$ be the space of all $f$ satisfying (1). Then $bC\sp n\sb{\lambda}\subset C\sb{\lambda}\sp{n+1}$ so that one obtains a subcomplex $(C\sp n\sb{\lambda},b)$ of the Hochschild complex on $(C\sp n({\frak A},{\frak A}\sp*),b)$. The cohomology of this complex is the ”cyclic cohomology” of ${\frak A}$- denoted by $H\sp n\sb{\lambda}({\frak A}).$ 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 $K\sp 0({\frak A})$ is described by a pair ($\phi$,${\bar \phi}$) of *-homomorphisms of ${\frak A}$ into ${\cal B}(H)$, $H$ a Hilbert space, such that $\phi$ (x)-${\bar \phi}$(x) is a compact operator for all $x\in {\frak A}$. If now ($\phi$,${\bar \phi}$) is even such that $q(x)=\phi (x)-{\bar \phi}(x)$ is in the Schatten class ${\cal C}\sp{p+1}(H)$ for all $x\in {\frak A}$, one can define $f(x\sb 0,...,x\sb p)=\text{Tr}(q(x\sb 0)...q(x\sb p))$ where $\text{Tr}$ is the usual trace ${\cal C}\sp 1(H)\to {\bbfC}$. One easily checks that $f\in C\sp p\sb{\lambda}({\frak A})$ if p is even and one can define the Chern character ch by $ch\sb p((\phi,{\bar \phi}))=[f]\in H\sp p\sb{\lambda}({\frak A})$. This construction gives rise naturally to an operator $S: H\sp p\sb{\lambda}\to H\sb{\lambda}\sp{p+2}$ which has the property that $ch\sb{p+2}((\phi,{\bar \phi}))$ (which is also defined since ${\cal C}\sp{p+1}\subset {\cal C}\sp{p+3})$ equals S ch${}\sb p((\phi,{\bar \phi})).$ One can now define $H\sp{even}({\frak A})=\lim\sb{\to}(H\sp 0\sb{\lambda}({\frak A})\to\sp{S}H\sp 2\sb{\lambda \quad}({\frak A})\to\sp{S}...)$ and $H\sp p({\frak A})$ the image of $H\sp p\sb{\lambda}({\frak A})$ in $H\sp{even}$ divided by the image of $H\sb{\lambda}\sp{p-2}({\frak A})$ (of course, the odd case is treated similarly). Connes shows that for ${\frak A}={\cal C}\sp{\infty}(M)$, $M$ a smooth compact manifold, one finds that $H\sp n({\frak A})$ is equal to the de Rham homology group $H\sb n(M,{\Bbb C})$. He also establishes a long exact sequence $$...\to H\sp{n+1}({\frak A},{\frak A}\sp*)\to H\sp n\sb{\lambda}({\frak A})\to\sp{S}H\sb{\lambda}\sp{n+2}({\frak A})\to H\sp{n+2}({\frak A},{\frak A}\sp*)\to...$$ connecting cyclic cohomology with Hochschild cohomology, and uses this sequence for instance to compute $H\sp*({\frak A})$ 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

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
19D55$K$-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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