Let $ \Gamma $ be a second countable locally compact group (for instance a countable discrete group).
As it is said here : https://en.wikipedia.org/wiki/Baum%E2%80%93Connes_conjecture , one can define the assembly map $ \mu $ concerning the Baum-Connes conjecture by : $$ \mu : RK_{*}^{ \Gamma } ( \underline{E \Gamma } ) \to K_{*} ( C_{r}^* ( \Gamma ) ) $$ from the equivariant $ K $ -homology with $ \Gamma $ -compact supports of the classifying space of proper actions $ \underline{E\Gamma} $ to the $ K $ -theory of the reduced $ C^* $ -algebra of $ \Gamma $ . The subscript index * can be $ 0 $ or $ 1 $.
I would like to know what is explicitely the definition of $ \mu $ ?
In some books, we say that $ \mu $ is defined by : $ \mu (Z,D) = \mathrm{Ind} ( D) $ without defining in reality $ \mathrm{Ind} (D) $ like in Alain Valette's book for instance. What is explicitely the definition of $ \mathrm{Ind} (D) $ ? In some books, i see that : $ \mathrm{Ind} (D) = \mathrm{dim} \ker D - \mathrm{dim} \ \mathrm{coker} D $ when for instance $ D $ is an elliptic partial differential operator, but here $ \mathrm{Ind} (D) = \mathrm{dim} \ker D - \mathrm{dim} \ \mathrm{coker} D $ is in $ \mathbb{Z} $ not in $ K_{*} ( C_{r}^* ( \Gamma ) ) $, no ?. In some electronic books like here : http://www.mmas.univ-metz.fr/~gnc/bibliographie/BaumConnes/Baum-Connes-Higson.pdf , page : $ 11 $ we define $ \mathrm{Ind} $ by : $ \mathrm{Ind} ( D ) = [ \ker D ] - [ \mathrm{coker} D ] $, so it's different from the first one : $ \mathrm{Ind} (D) = \mathrm{dim} \ker D - \mathrm{dim} \ \mathrm{coker} D $. So, i'm completely confused, what is in reality, the definition of the assembly map : $ \mu (Z,D) = \mathrm{Ind} ( D) $ appearing in the definition of the Baum Connes conjecture ?
Thanks in advance for your help.