2. NONCOMMUTATIV E SPACE S 5
2.1. Morit a equivalence . I n noncommutative geometry, isomor -
phisms o f C*-algebra s ar e to o restrictiv e t o provid e a goo d notio n o f
isomorphisms o f noncommutativ e spaces . Th e correc t notio n i s pro -
vided b y Morit a equivalenc e o f C*-algebras .
We have equivalent C*-algebra s A\ ~ A2 if there exists a bimodule
A4, which is a right Hilbert A\ modul e with an 4i-valued inner produc t
(',-)AH
a n ( i a kf t Hilber t ^-modul e wit h a n ^-valued inne r produc t
(•,
-)A2-
suc h tha t w e have:
W e obtain al l Ai a s the closur e of the spa n of
{(tub)*'• tub CM}.
V£i,£
2
,£3 E M we have
A\ an d A2 ac t o n A4 b y bounded operators ,
(a2€,a2€)Ai hzfi^OA! (ai€,aiQA2 ll
a
i||
2
(^0^2
for al l a\ E Au ^ 2 £ A2, £ M.
This notio n o f equivalenc e roughl y mean s tha t on e ca n transfe r
modules bac k an d fort h betwee n th e tw o algebras .
2.2. Th e tool s o f noncommutative geometry . Onc e one iden-
tifies i n a specific proble m a space that, b y its nature of quotient o f the
type describe d above , i s bes t describe d a s a noncommutativ e space ,
there i s a larg e se t o f wel l develope d technique s tha t on e ca n us e t o
compute invariant s an d extrac t essentia l informatio n fro m th e geome -
try. Th e following i s a list o f some such techniques, som e of which will
make their appearanc e i n the case s treated i n these notes .
Topologica l invariants: K-theor y
Hochschil d an d cycli c cohomolog y
Homotop y quotients , assembl y ma p (Baum-Connes )
Metri c structure : Dira c operator, spectra l triple s
Characteristi c classes , zeta function s
We wil l recal l th e necessar y notion s whe n needed . W e no w begi n
by taking a close r loo k a t th e analo g i n th e noncommutativ e worl d of
Riemannian geometry , which is provided by Connes' notion of spectra l
triples.
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