process

60 A. Bartels, S. Echterhoff, and W. Lück6.1 Desired equivariant homology theories. We will need the followingTheorem 6.1 (Construction of equivariant homology theories). Suppose that we aregiven a group and a ring R (with involution) or a C -algebra A respectively onwhich acts by structure preserving automorphisms. Then:(i) Associated to these data there are equivariant homology theories with values inZ-modules over the group H ‹ . I K R/H ‹ . I KH R/H ‹ . I Lh 1iR/;H ‹ . I Ktop /;A;l 1H ‹ . I Ktop A;r /;H ‹ . I Ktop A;m /;where in the case H ‹. I Ktop A;r/ we will have to impose the restriction to theinduction structure that a homomorphisms ˛ W .H; / ! .G; / over inducesa transformation ind˛ W Hn H .X; X 0/ ! Hn G.˛.X; X 0 // only if the kernel of theunderlying group homomorphism ˛ W H ! G acts with amenable isotropy onX n X 0 .(ii) If .G; / is a group over and H G is a subgroup, then there are for everyn 2 Z identificationsH H n .fgI K R/ Š H G n .G=H I K R/ Š K n .R Ì H/IH H n .fgI KH R/ Š H G n .G=H I KH R/ Š KH n .R Ì H/IH H nH H nH H nH H n.fgI Lh1iR/ Š Hn G 1i.G=H I LhR/ Š L h n1i .R Ì H/I.fgI Ktop / Š H G A;l 1 n .G=H I Ktop / Š KA;l 1 n .A Ì l 1 H/I.fgI KtopA;r / Š H n G .G=H I KtopA;r / Š K n.A Ì r H/I.fgI KtopA;m / Š H n G .G=H I KtopA;r / Š K n.A Ì m H/:Here H and G act on R and A respectively via the given -action, W G ! and the inclusion H G, K n .R Ì H/ is the algebraic K-theory of the twistedgroup ring R Ì H ,KH n .R Ì H/is the homotopy K-theory of the twisted groupring R Ì H , L hn1i .R Ì H/is the algebraic L-theory with decoration h 1i ofthe twisted group ring with involution R Ì H , K n .A Ì l 1 H/ is the topologicalK-theory of the crossed product Banach algebra A Ì l 1 H , K n .A Ì r H/ isthe topological K-theory of the reduced crossed product C -algebra A Ì r H ,and K n .A Ì m H/ is the topological K-theory of the maximal crossed productC -algebra A Ì m H ;