process

46 A. Bartels, S. Echterhoff, and W. Lückfor H and G with the map K n .Cr .˛//W K n.Cr .H // ! K n.Cr .G// provided that˛ has amenable kernel and hence Cr .˛/ is defined. So the Baum–Connes conjectureimplies that every group homomorphism ˛ W H ! G induces a group homomorphism˛ W K n .Cr .H // ! K n.Cr .G//, although there may be no C -homomorphismCr .H / ! C r .G/ induced by ˛. No such direct construction of ˛ is known in general.Here is another failure of the reduced group C -algebra. Let G be the colimit of thedirected system fG i j i 2 I g of groups (with not necessarily injective structure maps).Suppose that for every i 2 I and preimage H of a finite group under the canonical mapi W G i ! G the Baum–Connes conjecture for the maximal group C -algebra holds(This is for instance true by [22] if ker. i / has the Haagerup property). Thencolim i2I H G in .E F in.G i /I K topC;m / Š ! colim i2I H G inŠ! HGn .E F in .G/I K topC;m /.E i F in.G i /I K topC;m /is a composition of two isomorphisms. The first map is bijective by the TransitivityPrinciple 4.3, the second by Lemma 3.4 and Lemma 6.2. This implies that the followingcomposition is an isomorphismcolim i2I H G in .E F in.G i /I K topC;r / ! colim i2I H G in! H G n .E F in.G/I K topC;r /.E i F in.G i /I K topC;r /Namely, these two compositions are compatible with the passage from the maximal tothe reduced setting. This passage induces on the source and on the target isomorphismssince E F in .G i / and E F in .G/ have finite isotropy groups, for a finite group H we haveCr .H / D C m .H / and hence we can apply [13, Lemma 4.6]. Now assume furthermorethat the Baum–Connes conjecture for the reduced group C -algebra holds for G i foreach i 2 I and for G. Then we obtain an isomorphismcolim i2I K n .C r .G i// Š ! K n .C r .G//:Again it is in general not at all clear whether there exists such a map in the case, wherethe structure maps i W G i ! G do not have amenable kernels and hence do not inducemaps Cr .G i/ ! Cr .G/.These arguments do not apply to the Farrell–Jones conjecture or the Bost conjecture.Namely any group homomorphism ˛ W H ! G induces maps R Ì H ! R Ì G,A Ì l 1 H ! A Ì l 1 G, and A Ì m H ! A Ì m G for a ring R or a C -algebra A withstructure preserving G-action, where we equip R and A with the H -action comingfrom ˛. Moreover we will show for a directed system fG i j i 2 I g of groups (withnot necessarily injective structure maps) and G D colim i2I G i that there are canonical