4 1. OUVERTUR E

It i s important t o notice that, usually , the notion s of noncommuta -

tive geometry ar e "richer " tha n th e corresponding entries of the dictio-

nary on the commutative side. Fo r instance, as Connes discovered, non-

commutative measur e space s (vo n Neuman n algebras ) com e endowe d

with a natural tim e evolution which is trivial in the commutative case.

Similarly, a t th e leve l o f topology on e ofte n see s phenomen a tha t ar e

closer to rigid analytic geometry. Thi s is the case, for instance, with th e

noncommutative tor i T$, which alread y a t th e C*-algebr a leve l exhibit

moduli tha t behav e muc h like moduli o f one-dimensional comple x tor i

(elliptic curves ) i n the commutativ e case .

In th e contex t w e are goin g to discus s this richer structur e o f non -

commutative space s i s crucial , a s i t permit s u s t o us e tool s lik e C* -

algebras (topology ) t o stud y th e propertie s o f mor e rigi d space s lik e

algebraic o r arithmeti c varieties .

2. Noncommutativ e space s

The wa y t o assig n th e algebr a o f coordinate s t o a quotien t spac e

X = Yf ~ ca n b e explained i n a short sloga n a s follows :

• Function s o n Y wit h f(a) = /(& ) fo r a ^ b. Poor !

• Function s /

a

& on the graph of the equivalence relation. Good !

The secon d descriptio n lead s to a noncommutativ e algebra , a s th e

product, determine d b y th e groupoi d la w o f th e equivalenc e relation ,

has the for m o f a convolution produc t (lik e the produc t o f matrices).

For sufficientl y nice quotients , eve n thoug h th e tw o notion s ar e

not th e same , the y ar e relate d b y Morit a equivalence , whic h i s th e

suitable notion of "isomorphism " betwee n noncommutative spaces. Fo r

more genera l quotients , however , th e tw o notion s trul y diffe r an d th e

second on e i s the onl y on e tha t allow s on e t o continu e t o mak e sens e

of geometry o n the quotien t space .

A ver y simpl e exampl e illustratin g th e abov e situatio n i s th e fol -

lowing (cf. [25]) . Conside r th e topologica l spac e Y = [0,1] x {0,1}

with th e equivalenc e relatio n (x , 0) ~ (x

7

1) fo r x E (0,1). B y the first

method on e onl y obtain s constan t function s C , whil e b y th e secon d

method on e obtain s

{/ G C([0,1])g M2(C) : /(0 ) an d /(l ) diagona l }

which is a n interestin g nontrivia l algebra .

The idea of preserving the information on the structure of the equiv-

alence relatio n i n th e descriptio n o f quotien t space s ha s analog s i n

Grothendieck's theor y o f stacks in algebrai c geometry .