The Bost conjecture and proper Banach algebras (Q1946285)

Let \(\mathcal{G}\) be a locally compact Hausdorff groupoid that is equipped with a Haar system and for which there exists a locally compact classifying space \(\underline{E}\mathcal{G}\) for proper actions of \(\mathcal{G}\). Using the bivariant \(KK\)-theory for Banach algebras defined by \textit{V. Lafforgue} [Invent. Math. 149, No.~1, 1--95 (2002; Zbl 1084.19003)], one can define for each nonnegative integer \(n\) and any \(\mathcal{G}\)-Banach algebra \(B\) an abelian group \(K_n^{\text{top, ban}}(\mathcal{G}, B)\). For an unconditional completion (this term is defined in the paper under review) \(\mathcal{A}(\mathcal{G}, B)\) of \(\mathcal{C}_c(\mathcal{G}, B)\), the author defines a Bost assembly map \(\mu_{\mathcal{A}}^{\mathcal{B}}\) from \(K_n^{\text{top, ban}}(\mathcal{G}, B)\) to \(K_n(\mathcal{A}(\mathcal{G}, B))\). The Banach algebraic version of the Bost conjecture asserts that \(\mu_{\mathcal{A}}^{\mathcal{B}}\) is an isomorphism. In this paper, the author defines a notion of a proper \(\mathcal{G}\)-Banach algebra and proves that under mild conditions on \(\mathcal{A}\), the Bost assembly map \(\mu_{\mathcal{A}}^{\mathcal{B}}\) is split surjective for any non-degenerate proper \(\mathcal{G}\)-Banach algebra \(B\). Furthermore, the splitting is natural in \(B\). The author proves the result by first establishing it when \(\mathcal{G}\) is proper and \(\underline{E}\mathcal{G}\slash\mathcal{G}\) is compact; to do this, he employs a generalized Green-Julg theorem that appears in another paper of his [J. Noncommut. Geom. 7, No. 1, 149--190 (2013; Zbl 1273.43003)].