International Congress of Mathematicians

804 V. Lafforguefree group and called (RD) (for rapid decay) by Jolissaint ([23]). In case a),b) Ghas property (RD) : this is due to Haagerup for free groups ([16]), Jolissant for"geometric hyperbolic groups", de la Harpe for general hyperbolic groups ([18]),Ramagge, Robertson and Steger for SL% of a non-archimedian local field ([47]), theauthor for SL 3 (R) and SL 3 (C) ([39]), Chatterji for SL 3 (M) and E 6{ _ 26) ([10]), andthe remark that it holds for products is due to Ramagge, Robertson and Steger([47]) in a particular case, and independantly to Chatterji ([10]) and Talbi ([50])in general. A discrete group G has property (RD) if there is a lenght function£ : G —¥ R+ (i.e. a function satisfying i(g^v)= 1(g) and £(gh) < 1(g) + 1(h) forany g,h £ G) such that for s £ R+ big enough, the completion H S (G) of CG forthe norm || E f(g)e g \\ H *(G) = II EC 1 + %)) s /(^KIIL^G) is contained in G r * ed (G).Then, for s big enough, H S (G) is a Banach algebra and an involutive subalgebra ofC* ed (G) and is dense and stable under holomorphic functional calculus ([23, 39]); itis obvious that H S (G) is an unconditional completion of CG.As a consequence of this result the Baum-Connes conjecture has been provenfor all almost connected groups by Chabert, Echterhoff and Nest ([9]).1.7. Trying to push the method furtherIn order to prove new cases of the surjectivity of the Baum-Connes map (whenthe injectivity is proven and the 7 element exists) we should look for a densesubalgebra -4(G) of C* ed (G) that is stable under holomorphic functional calculusand a homotopy between 7 and 1 through (perhaps special kind of) elements ofE bAn (C, C) which all give a map K„,(A(Gj) —¥ K*(C* ed (Gj) by the descent construction.Thanks to the discussion in subsection 1. a necessary condition for this is thatfor any (E, n, T) in the homotopy between 7 and 1, for any x £ E and £ £ Cc(E, C),the Schur multiplication by the matrix coefficient g H> Ç(ir(g)(xj) is bounded fromA(G) to C* ed (G) and has norm < ||X||E:||£||£ r {E,Q- So we should first look for ahomotopy between 7 and 1 such that the fewest possible matrix coefficients appear.For groups acting properly on buildings, this homotopy can be shown to exist. Theproblem for general discrete groups properly acting on buildings is to find a subalgebrav4(G) of C* ed (G) that is stable under holomorphic functional calculus andsatisfies the condition with respect to these matrix coefficients. The first step (thecrucial one I think) should be to find a subalgebra -4(G) of C* ed (G) that is stableunder holomorphic functional calculus and satisfies the following condition : thereis a integer n, a distance d on the building and a point xo on the building such thatthe Schur product by the characteristic function of {g £ G,d(xo,gxo) < r} fromA(G) to C* ed (G) has norm less than (1 + r) n , for any r £ R+.1.8. The Baum-Connes conjecture with coefficientsLet G be a second countable, locally compact group and A a G-Banach algebra(i.e. a Banach algebra on which G acts continuously by isometric automorphismsg : a >-¥ g (a)). The space C C (G,A) of A-valued continuous compactly supportedfunctions on G is endowed with the following convolution product : f * f'(g) =JG fWh(f'(h^19J)dhand the completion L 1 (G, A) of C C (G, A) for the norm ||/|| =