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
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
( / i * /
) ( 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. Spherical case. Consider the action of PSL2CZ) on P
(C) by nomo-
graphics. Note that
(C) = H
( R ) , H
: = C \ R = H
U H " ,
= H is the upper half plane in C, H~ is the lower half plane, and P
(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
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
( R ) - + P
( R )
and consider the C*—algebra C*(G) modulo Morita equivalence. The latter is de-
clared to be, by definition, the "quotient" P
(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
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