Twisted coarse Baum–Connes conjecture with coefficients

Let $X$ be a metric space with bounded geometry, let $P_d(X)$ be its Rips complex at scale $d$, and let $\Gamma(X,\mathcal A)$ be a coarse $X$-algebra. The twisted assembly map is

$$\mu_{\Gamma(X,\mathcal A)}:\lim_{d\to\infty}K_*(C^*_{L,\Gamma(X,\mathcal A)}(P_d(X),\mathcal A))\to K_*(C^*_{\Gamma(X,\mathcal A)}(X,\mathcal A)).$$

Here $C^*_{L,\Gamma(X,\mathcal A)}(P_d(X),\mathcal A)$ is the twisted localization algebra and $C^*_{\Gamma(X,\mathcal A)}(X,\mathcal A)$ is the twisted Roe algebra with coefficients in $\Gamma(X,\mathcal A)$. Twisted coarse Baum–Connes conjecture with coefficients. For every coarse $X$-algebra $\Gamma(X,\mathcal A)$, the map $\mu_{\Gamma(X,\mathcal A)}$ is an isomorphism. The paper proves this conjecture for spaces with suitable coarse-fibration structures, including cases whose base and fiber satisfy the corresponding conjecture; its general status is not resolved here.