0.3. COMPARISON WITH OTHER THEORIES xxvii

shall examine some examples. Our discussion will inevitably be extremely super-

ficial and not entirely precise; for precise, in-depth presentations we refer to [36],

[99], [101].

Connes' theory is formulated in the context of functional analysis and, although

some parts of it have a more algebro-geometric flavor [127] [99], our presentation

will follow

Connes7

original approach. As already mentioned the starting point in

this theory is often a groupoid

Q = (X,X, o-i,cr2,^, L,C)

in some "geometric" category, for instance the category of smooth manifolds. Write

7i72 = ^(71,72)- The first step is to attach to Q a C*—algebra, C*((/), called the

convolution algebra of Q. Naively C*(Q) might be thought of as (a norm completion

of) some algebra of complex valued functions f on X with multiplication given by

"convolution":

( / i * /

2

) ( 7 ) = E /i(7i)/2(72).

71 72 = 7

As it is this cannot work because the sums involved are usually infinite. So one

replaces, according to the context, sums by integrals, functions by "densities", etc.

The next step in Connes' theory is the realization that "non-commutative

spaces" should not be defined as C*— algebras but rather as Morita equivalence

classes of C*—algebras. Two C*—algebras A and B are Morita equivalent if there

exists a bimodule A^B with certain extra data that allow one to "transfer" modules

between A and B. So, roughly speaking, one defines the non-commutative space

attached to the groupoid Q as the C*—algebra C*(Q) up to Morita equivalence.

At this point one is faced with the "topological and geometrical study of non-

commutative spaces"; this constitutes the heart of the theory and, for this purpose,

a whole spectrum of techniques is brought into the picture such as: spectral theory,

index theory, K—theory, etc.

In what follows we examine a number of examples in non-commutative geom-

etry and compare them with the spherical, flat, and hyperbolic examples we are

able to treat in our theory.

0.3.5.1. Spherical case. Consider the action of PSL2CZ) on P

1

(C) by nomo-

graphics. Note that

P

1

(C) = H

±

U P

1

( R ) , H

±

: = C \ R = H

+

U H " ,

H

+

= H is the upper half plane in C, H~ is the lower half plane, and P

1

(R) is the

real projective line. The action of PSL2{Z) on H

±

is properly discontinuous and

the quotient exists in the category of Riemann surfaces; it is the union of 2 copies

of the modular curve

H/PSrI/2(Z)

parameterizing elliptic curves over the complex

numbers. However the action of PSL^i^) on the "stratum" P 1 (R) is not properly

discontinuous and actually the categorical quotient in the category of manifolds

reduces to a point. What can be done in non-commutative geometry is to consider

the groupoid Q attached to the action

PSL2(Z) x P

1

( R ) - + P

1

( R )

and consider the C*—algebra C*(G) modulo Morita equivalence. The latter is de-

clared to be, by definition, the "quotient" P

1

(R)/P5L

2

(Z) and can be interpreted,

in a natural way as a compactification of the modular curve H/PSX2(Z); cf. the

remarks in the next example. The non-commutative space P1(R)/P5L2(Z) has