*Approximation properties for groups and C*-algebras.*

It is classical result in Fourier analysis, that the Fourier series
of a continuous function may fail to converge uniformly or even
pointwise to the given function. However if one use a summation
method as e.g. convergence in Cesaro mean, one actually gets uniform
convergence of the Fourier series. This result can easily be generalized
first to all abelian LC (= locally compact) groups, and next to
all amenable (LC) groups, where in the non-abelian case, the continuous
functions on dual group G^ should be replaced by the reduced group
C*-algebra of G.

In 1994 Jon Kraus and I introduced a new approximation property
(AP) for locally compact groups. The groups having (AP) is the largest
class of LC-groups for which a generalized Cesaro mean convergence
theorem can hold. The group SL(2,R) has this property, but it was
only proven recently by Vincent Lafforgue and Mikael de la Salle,
that SL(n,R) fails to have (AP) for n = 3,4,... In a joint work
with Tim de Laat we extend their result by proving that Sp(2,R)
and more generally all simple connected Lie groups of real rank
>=2 and with finite center do not have the (AP).

In the talk I will give an introduction to amenabily, weak amenability
and the property (AP) for locally compact groups, and the corresponding
properties for C*-algebras will also be discussed. Weak amenability
is another approximation property for LC-groups.

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