Main discovery. For every countable discrete group Γ and every Γ–C*-algebra A, the map π_* induced by the natural Γ‑equivariant map π: EΓ → ĒΓ is injective after tensoring with the rationals. Concretely, π_*: KK^Γ_*(EΓ, A) ⊗_Z Q → KK^Γ_*(ĒΓ, A) ⊗_Z Q is injective (Theorem 1.2). (KK denotes Kasparov’s equivariant KK-theory.)

How the authors prove it. They give an elementary proof of rational injectivity that avoids II_1 factors and does not use equivariant KK-theory with real coefficients.

Key ingredients and steps. 1) Rational idempotents. For every finite proper Γ‑CW complex X the authors construct a canonical idempotent e_X ∈ K^0_Γ(X) ⊗ Q using the equivariant Chern character of Lück–Oliver. The idempotents are natural in X and satisfy e_{Γ/H} = ε_H, where ε_H = |H|^{-1}[C_H] in R(H) ⊗ Q for finite subgroups H.

2) Module action. K^0_Γ(X) acts on KK^Γ_n(C_0(X), B) via the Kasparov product, making these KK-groups into K^0_Γ(X)-modules. The action extends rationally to K^0_Γ(X) ⊗ Q on KK-groups ⊗ Q.

3) Reduction to finite X and separable coefficients. Choosing countable Γ‑CW models and passing to inductive limits over finite Γ‑subcomplexes and separable Γ‑invariant coefficient subalgebras reduces the global injectivity problem to the following target. (p_X)_*: KK^Γ_*(EΓ × X, B) ⊗ Q → KK^Γ_*(X, B) ⊗ Q One must prove injectivity of this map for each finite proper Γ‑CW complex X and each separable Γ–C*-algebra B.

4) Orbit case and explicit splitting. When X = Γ/H with H finite, the authors define compression and transfer maps and compute their composite. On finite H‑subcomplexes Δ this composite equals multiplication by the class of the trivial bundle Δ×C_H with diagonal H‑action, so (tr_Δ ◦ r_Δ)(x) = [Δ×C_H]·x. Assembled globally, this yields J_E ◦ R_E = |H|·id. Using the identity [V] = |H| e_X (where [V] corresponds to the regular representation C_H), they construct an explicit rational splitting σ, which gives injectivity in the orbit case.