There is an interpretation of the Baum-Connes assembly map as the index of a $C^*_r(G)$-linear operator in case $G$ is discrete and torsion-free. The group $K_*(BG)$ can be represented by triples $(M, E, f)$, where $M$ is a closed spin$^c$-manifold, $E$ is a complex vector bundle and $g \colon M \to BG$ is a continuous map with respect to an equivalence relation given in Section 5 here.

Let $\mathcal{V} = \widetilde{M} \times_{\lambda} C^*_r(\pi_1(M))$ be the bundle of right Hilbert $C^*_r(\pi_1(M))$-modules over $M$, where $\lambda \colon \pi_1(M) \to Aut(C^*_r(\pi_1(M)))$ is the left multiplication with elements of $\pi_1(M)$. Let $D$ be the Dirac operator on $M$. We can modify $D$ to act on smooth sections of the bundle $S \otimes E \otimes \mathcal{V}$ over $M$, where $S$ is the spinor bundle. Mishchenko and Fomenko have developed a theory of how to associate and index to these $A$-linear operators for some $C^*$-algebra $A$. This can be found here. The map $[M,E,f] \mapsto [ind(D^{E \otimes \mathcal{V}}_+)]$ is the assembly map.

They also prove an index theorem for these operators, which involves a topological index on the right hand side. For this you first need a Chern character, which can be obtained for nice enough $C^*$-algebras from the Künneth theorem for $K$-theory, i.e. we have isomorphisms $$ K_0(C(M,A)) \otimes \mathbb{R} = K_0(C(M) \otimes A) \otimes \mathbb{R} \cong \left(K_0(C(M)) \otimes K_0(A) \oplus K_1(C(M)) \otimes K_1(A) \right) \otimes \mathbb{R} $$ which yields a map $K_0(C(M,A)) \to K_0(C(M)) \otimes K_0(A) \otimes \mathbb{R}$ by projection to one summand and finally a Chern character $$ ch \colon K_0(C(M,A)) \to H^{\rm even}(M; K_0(A) \otimes \mathbb{R}) $$ by applying the usual Chern character to $K_0(C(M))$. The group $K_0(C(M,A))$ is the natural home for (formal differences of) isomorphism classes of Hilbert $A$-module bundles over $M$. In particular $E \otimes \mathcal{V}$ represents an element in $K_0(C(M,C^*_r(\pi_1(M)))$. A corollary of the Mishchenko-Fomenko index theorem is now that $$ {\rm ind}(D^{E \otimes \mathcal{V}}_+) = \int_M \hat{A}(M)\cdot ch(E \otimes \mathcal{V}) $$ and there you have your topological index (if I have done everything correct). I glossed over some things here, e.g. the grading on $S$. I also recommend the notes by Higson about the Baum Connes conjecture. (I recommend reading everything by Higson. He does a marvelous job at explaining things!)

There is an interpretation of the Baum-Connes assembly map as the index of a $C^*_r(G)$-linear operator in case $G$ is discrete and torsion-free. The group $K_*(BG)$ can be represented by triples $(M, E, f)$, where $M$ is a closed spin$^c$-manifold, $E$ is a complex vector bundle and $g \colon M \to BG$ is a continuous map with respect to an equivalence relation given in Section 5 here.

Let $\mathcal{V} = \widetilde{M} \times_{\lambda} C^*_r(\pi_1(M))$ be the bundle of right Hilbert $C^*_r(\pi_1(M))$-modules over $M$, where $\lambda \colon \pi_1(M) \to Aut(C^*_r(\pi_1(M)))$ (module automorphisms) is the left multiplication with elements of $\pi_1(M)$. Let $D$ be the Dirac operator on $M$. We can modify $D$ to act on smooth sections of the bundle $S \otimes E \otimes \mathcal{V}$ over $M$, where $S$ is the spinor bundle. Mishchenko and Fomenko have developed a theory of how to associate and index to these $A$-linear operators for some $C^*$-algebra $A$. This can be found here. The map $[M,E,f] \mapsto [ind(D^{E \otimes \mathcal{V}}_+)]$ is the assembly map.

They also prove an index theorem for these operators, which involves a topological index on the right hand side. For this you first need a Chern character, which can be obtained for nice enough $C^*$-algebras from the Künneth theorem for $K$-theory, i.e. we have isomorphisms $$ K_0(C(M,A)) \otimes \mathbb{R} = K_0(C(M) \otimes A) \otimes \mathbb{R} \cong \left(K_0(C(M)) \otimes K_0(A) \oplus K_1(C(M)) \otimes K_1(A) \right) \otimes \mathbb{R} $$ which yields a map $K_0(C(M,A)) \to K_0(C(M)) \otimes K_0(A) \otimes \mathbb{R}$ by projection to one summand and finally a Chern character $$ ch \colon K_0(C(M,A)) \to H^{\rm even}(M; K_0(A) \otimes \mathbb{R}) $$ by applying the usual Chern character to $K_0(C(M))$. The group $K_0(C(M,A))$ is the natural home for (formal differences of) isomorphism classes of Hilbert $A$-module bundles over $M$. In particular $E \otimes \mathcal{V}$ represents an element in $K_0(C(M,C^*_r(\pi_1(M)))$. A corollary of the Mishchenko-Fomenko index theorem is now that $$ {\rm ind}(D^{E \otimes \mathcal{V}}_+) = \int_M \hat{A}(M)\cdot ch(E \otimes \mathcal{V}) $$ and there you have your topological index (if I have done everything correct). I glossed over some things here, e.g. the grading on $S$. I also recommend the notes by Higson about the Baum Connes conjecture. (I recommend reading everything by Higson. He does a marvelous job at explaining things!)