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 /
& 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
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 .
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