Baum-Connes-like "conjecture" for $l^p$-spaces Let $G$ be a (discrete) group. For the Baum-Connes conjecture, one looks at the reduced group $C^{\star}$-algebra: Look at the Hilbert space $l^2(G)$ and the representation of $G$ on this Hilbert space given by left multiplication. The norm-closure of the resulting $\mathbb{C}G$-representation in $B(l^2(G))$ is the reduced group $C^*$-algebra.
For any $p \geq 1$, we can pretty much do the same: Look at $l^p(G)$. We still have a representation of $G$ on $B(l^p(G))$ by left multiplication and hence obtain a kind of reduced Banach group algebra for $l^p$; lets call it $B^p(G)$.
There also should be an assembly map
$K_*(E_{Fin}G) \rightarrow K_*(B^p(G))$
as in the Baum-Connes conjecture. For $p = 1$, we have $B^1(G) = l^1(G)$ and we obtain the Bost assembly map. I have no reason to believe that for arbitrary p such an assembly map might be an isomorphism, but was wondering whether such group Banach algebras, and maybe even the assembly maps, have been considered anywhere in the literature.  
 A: It is likely that in cases where I proved Baum-Connes without coefficients (i.e. reductive groups over local fields and some discrete groups with RD), some variant of the Schwartz or Jolissaint algebra will be dense and stable under functional calculus in the algebra you call $B^p(G)$. This would imply the BC conjecture for it. 
In the case of Schwartz algebras, some arguments like this (with almost the right L^p estimates)
are given in the last section of my paper in Inventiones. 
The big limitation of this $L^p$ variant of the Baum-Connes conjecture is that when you consider coefficients in a $G$-$C^*$-algebra A, $L^p(G,A)$ 
can be defined only in a naive way and cannot be a  $A$-Hilbert module as in the case where p=2. 
A: There is a version of $KK$-theory for Banach algebras, which was developed by Lafforgue. There also is a paper titled Banach KK-theory and the Baum-Connes conjecture, which is probably relevant for this question. I think this is a survey of K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes.
