An elementary proof of rational injectivity in equivariant KK-theory

Antonini, Azzali and Skandalis proved that, for every countable discrete group $\mathrm{\Gamma }$, the canonical map $E\mathrm{\Gamma }\to \underset{‾}{E}\mathrm{\Gamma }$ induces a split injection in equivariant $KK$-theory with real coefficients. In this paper, we give a more elementary proof of the corresponding rational injectivity result in ordinary equivariant $KK$-theory without using ${\mathrm{I}\mathrm{I}}_1$ factors.