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Rectified Baum–Guentner–Willett (BGW) conjecture for admissible exact crossed products

Show that for i in {0,1}, the BGW assembly map μ^{BGW}_{i,}: RK_i(Γ, X) → K_i(C(X) ⋊_{} Γ) is an isomorphism, where μ^{BGW}_{i,} is defined as the maximal assembly map composed with the canonical map induced by the minimal exact, Morita compatible crossed-product completion.

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Background

To address failures of the reduced Baum–Connes conjecture in non-exact settings, Baum–Guentner–Willett introduced a rectified version using the minimal exact and Morita compatible crossed product. The paper states this rectified conjecture explicitly, which seeks to recover an isomorphism theorem at the level of K-theory using the BGW completion.

References

This rectified conjecture can be stated as follows. For $i\in 2$, the composite morphism $$ \mu{BGW}{i,}: {RK_i (\Gamma, X)} \longrightarrow K_i (C_{}*()), $$ of $\mu{\max}_{i,}$ with the natural morphism $K_i(C_{\max}*())\rightarrow K_i(C*_{ } ())$, is an isomorphism.

— Admissible Higson-Roe sequences for transformation groupoids (2411.00182 - Benameur et al., 31 Oct 2024) in Subsection 4.1 (Review of the BC assembly map), Conjecture [rectified BC conjecture]