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.