process

Inheritance of isomorphism conjectures under colimits 61(iii) Let W 0 ! 1 be a group homomorphism. Let R be a ring (with involution)and A be a C -algebra on which 1 acts by structure preserving automorphisms.Let .G; / be a group over 0 . Then in all cases the evaluation at .G; / of theequivariant homology theory over 0 associated to R or A respectivelyagrees with the evaluation at .G; ı / of the equivariant homology theory over 1 associated to R or A respectively.(iv) Suppose the group acts on the rings (with involution) R and S or on theC -algebras A and B respectively by structure preserving automorphisms. Let W R ! S or W A ! B be a -equivariant homomorphism of rings (withinvolution) or C -algebras respectively. Then induces natural transformationsof homology theories over ‹ W H ‹ . I K R/ ! H ‹ . I K S/I ‹ W H ‹ . I KH R/ ! H ‹ . I KH S/I ‹ W H ‹ . I Lh 1iR/ ! H ‹ 1i. I LhS/I ‹ W H ‹ . I Ktop / ! H ‹ A;l 1 . I Ktop /IB;l 1 ‹ W H ‹ . I Ktop A;r / ! H ‹ . I Ktop B;r /I ‹ W H ‹ . I Ktop A;m / ! H ‹ . I Ktop B;m /:They are compatible with the identifications appearing in assertion (ii).(v) Let act on the C -algebra A by structure preserving automorphisms. Wecan consider A also as a ring with structure preserving G-action. Then thereare natural transformations of equivariant homology theories with values in Z-modules over H ‹ . I K A/ ! H ‹ . I KH A/ ! H ‹ . I Ktop /A;l 1! H ‹ . I Ktop A;m / ! H ‹ . I Ktop A;r /:They are compatible with the identifications appearing in assertion (ii).6.2 (Strong) continuity. Next we want to showLemma 6.2. Suppose that we are given a group and a ring R (with involution) or aC -algebra A respectively on which G acts by structure preserving automorphisms.Then the homology theories with values in Z-modules over H ‹ . I K R/; H ‹ . I KH R/; H ‹ 1i. I LhR/; H ‹ . I Ktop /; and H ‹ A;l 1 . I Ktop A;m /(see Theorem 6.1) are strongly continuous in the sense of Definition 3.3, whereasH ‹ . I Ktop A;r /is only continuous.