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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.