A generalised Green–Julg theorem for proper groupoids and Banach algebras

Abstract

The Green–Julg theorem states that ${\mathrm{K}}_0^G(B)\cong {\mathrm{K}}_0({\mathrm{L}}^1(G,B))$ for every compact group $G$ and every $G$-$C^{\ast }$-algebra B. We formulate a generalisation of this result to proper groupoids and Banach algebras. This is a key result in the proof that the Bost assembly map is surjective for proper Banach algebra coefficients.