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The study of idempotent elements in group algebras (or, more
generally, the study of classes in the K-theory of such algebras)
originates from geometric and analytic considerations. For example,
C.T.C. Wall [72] has shown that the problem of deciding whether a
?nitely dominated space with fundamental group? is homotopy
equivalent to a ?nite CW-complex leads naturally to the study of a
certain class in the reduced K-theoryK (Z?) of the group ringZ?. 0
As another example, consider a discrete groupG which acts freely,
properly discontinuously, cocompactly and isometrically on a
Riemannian manifold. Then, following A. Connes and H. Moscovici
[16], the index of an invariant 0th-order elliptic
pseudo-di?erential operator is de?ned as an element in the ? ? K
-group of the reduced groupC -algebraCG. 0 r
Theidempotentconjecture(alsoknownasthegeneralizedKadisonconjec- ? ?
ture) asserts that the reduced groupC -algebraCG of a discrete
torsion-free r groupG has no idempotents =0,1; this claim is known
to be a consequence of a far-reaching conjecture of P. Baum and A.
Connes [6]. Alternatively, one
mayapproachtheidempotentconjectureasanassertionabouttheconnect-
ness of a non-commutative space;ifG is a discrete torsion-free
abelian group ? thenCG is the algebra of continuous complex-valued
functions on the dual r
This interesting collection of up-to-date survey articles on
various topics of current mathematical research presents extended
versions of the plenary talks given by important Greek
mathematicians at the congress held in Athens, Greece, on occasion
of the celebration for the 100 years of the Hellenic Mathematical
Society.
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