process

Inheritance of isomorphism conjectures under colimits 63(ii) Given A i 2 M.m;nI RÌG i / and A j 2 M.m;nI RÌG j / with i .A i / D j .A j /,there exists k 2 I with i;j k and i;k .A i / D j;k .A j /.An element ŒA in L h 0.R Ì G/ is represented by a quadratic form on a finitelygenerated free R Ì G-module, i.e., a matrix A 2 GL n .R Ì G/ for which there existsa matrix B 2 M.n;nI R Ì G/ with A D B C B , where B is given by transposingthe matrix B and applying the involution of R elementwise. Fix such a choice ofa matrix B. Choose i 2 I and B i 2 M.n;nI R Ì G i / with i .B i / D B. Theni.B i C Bi / D A is invertible. Hence we can find j 2 I with i j such thatA j WD i;j .B i C Bj / is invertible. Put B j D i;j .B i /. Then A j D B j C Bj andj .A j / D A. Hence A j defines an element ŒA j 2 L h n .R Ì G j / which is mapped toŒA under the homomorphism L h n .R Ì G j / ! L n .R Ì G/ induced by j . Hence themap ! 0 of (6.2) is surjective.Next we deal with H ‹ . I Ktop A;l 1 /. We have to show for every directed systems ofgroups fG i j i 2 I g with G D colim i2I G i together with a map W G ! that thecanonical mapcolim i2I K n .A Ì l 1 G i / ! K n .A Ì l 1 G/is bijective for all n 2 Z. Since topological K-theory is a continuous functor, it sufficesto show that the colimit (or sometimes also called inductive limit) of the system ofBanach algebras fA Ì l 1 G i j i 2 I g in the category of Banach algebras with normdecreasing homomorphisms is A Ì l 1 G. So we have to show that for any Banachalgebra B and any system of (norm deceasing) homomorphisms of Banach algebras˛i W AÌ l 1 G i ! B compatible with the structure maps AÌ l 1 i;j W AÌ l 1 G i ! AÌ l 1 G jthere exists precisely one homomorphism of Banach algebras ˛ W A Ì l 1 G ! B withthe property that its composition with the structure map AÌ l 1 i W AÌ l 1 G i ! AÌ l 1 Gis ˛i for i 2 I .It is easy to see that in the category of C-algebras the colimit of the system fAÌG i ji 2 I g is A Ì G with structure maps A Ì i W A Ì G i ! A Ì G. Hence the restrictionsof the homomorphisms ˛i to the subalgebras A Ì G i yields a homomorphism of centralC-algebras ˛0 W A Ì G ! B uniquely determined by the property that the compositionof ˛0 with the structure map A Ì i W A Ì G i ! A Ì G is ˛ij AÌGi for i 2 I .If˛ exists,its restriction to the dense subalgebra A Ì G has to be ˛0. Hence ˛ is unique if it exists.Of course we want to define ˛ to be the extension of ˛0 to the completion A Ì l 1 G ofA Ì G with respect to the l 1 -norm. So it remains to show that ˛0 W A Ì G ! B is normdecreasing. Consider an element u 2 A Ì G which is given by a finite formal sumu D P g2F a g g, where F G is some finite subset of G and a g 2 A for g 2 F .Wecan choose an index j 2 I and a finite set F 0 G j such that j j F 0 W F 0 ! F is oneto-one.For g 2 F let g 0 2 F 0 denote the inverse image of g under this map. Considerthe element v D P g 0 2F 0 a g g 0 in A Ì G j . By construction we have A Ì j .v/ D uand kvk Dkuk D P niD1 ka ik. We concludek˛0.u/kDk˛0 ı .A Ì j /.v/k Dk˛j .v/k kvk Dkuk:The proof for H ‹. I Ktop A;m/ follows similarly, using the fact that by definition of thenorm on AÌ m G every -homomorphism of AÌG into a C -algebra B extends uniquely