By Charles A. Weibel

A portrait of the topic of homological algebra because it exists at the present time

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H n−1 (M ; OM ). They play an important role in our recent work about relating The second author is supported by the Fonds zur F¨ orderung der wissenschaftlichen Forschung (Austrian Science Fund), project number P14195-MAT. 38 D. Burghelea and S. Haller the topology of non-simply connected manifolds to the complex geometry/analysis of the variety of complex representations of their fundamental group. Both concepts existed in literature prior to our work, cf. [1] and [14]. We have extended, generalized and Poincar´e dualized them because of our needs, cf.

Fourier (Grenoble) 24 (1974), no. 3, xiv, 171–217. [7] A. Grothendieck, Produits tensoriels topologiques et espaces nucl´eaires. Mem. Amer. Math. Soc. (1955), no. 16, 140 pp. [8] U. Haagerup, An example of a non-nuclear C ∗ -algebra which has the metric approximation property. Inventiones Math. 50 (1979), 279–293. Approximation Properties 35 [9] U. Haagerup, J. Kraus, Approximation properties for group C ∗ -algebras and group von Neumann algebras. Trans. Amer. Math. Soc. 344 (1994), no. 2, 667–699.

If u ∈ M A(Γ) then u is a bounded continuous function and mu is a bounded operator on the space A(Γ). We say that u is a completely bounded multiplier if and only if the operator mu is completely bounded. The set M0 (A(Γ)) of completely bounded multipliers is equipped with the norm u M0 A(Γ) which we shall call the multiplier norm. = mu cb , 28 J. A. Niblo By analogy with Leptin’s result we have the following definition of weak amenability. 5. We say that a group Γ is weakly amenable iff A(Γ) has an approximate identity that is bounded in the multiplier norm.

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