The Bost conjecture and proper Banach algebras

Abstract

For proper Banach algebra coefficients, the Bost assembly map is split surjective and the right-hand side of the conjecture does not depend on the choice of the unconditional completion involved. These results hold for a large class of unconditional completions.

Expanders are counterexamples to the $\ell^p$ coarse Baum–Connes conjecture

“…[3, 8-10, 16, 21-24, 28, 38, 46-48]), the study of assembly maps involving L p operator algebras contributes to our general understanding of the Ktheory of some of these algebras. We also note that other assembly maps involving L p operator algebras have recently been considered in [8,16], as well as in unpublished work of Kasparov-Yu. In similar spirit, the Bost conjecture [35,44,45,56] asks whether the Baum-Connes-type assembly map into the K-theory of the Banach algebra L 1 .G/ is an isomorphism for a locally compact group G.…”

Expanders are counterexamples to the $\ell^p$ coarse Baum-Connes conjecture

“…We also note that other assembly maps involving L p operator algebras have recently been considered in [4,8], as well as in unpublished work of Kasparov-Yu. In similar spirit, the Bost conjecture [24,29,30,38] asks whether the Baum-Connes-type assembly map into the K -theory of the Banach algebra L 1 (G) is an isomorphism for a locally compact group G.…”

Groupoids and Hermitian Banach *-algebras

“…This map was shown to be an isomorphism when the group Γ belongs to a large class C ′ , called the Lafforgue class, which includes hyperbolic groups and semi-simple real Lie groups. The assembly map for unconditional completions of groupoids has been also studied by Paravicini [34]. There is then the question of when the -groups of * (Γ) and A(Γ) are isomorphic.…”