  Let $X$ be a discrete metric space with bounded geometry. We show that if $X$
admits an "A-by-CE coarse fibration", then the canonical quotient map $\lambda:
C^*_{\max}(X)\to C^*(X)$ from the maximal Roe algebra to the Roe algebra of
$X$, and the canonical quotient map $\lambda: C^*_{u, \max}(X)\to C^*_u(X)$
from the maximal uniform Roe algebra to the uniform Roe algebra of $X$, induce
isomorphisms on $K$-theory. A typical example of such a space arises from a
sequence of group extensions $\{1\to N_n\to G_n\to Q_n\to 1\}$ such that the
sequence $\{N_n\}$ has Yu's property A, and the sequence $\{Q_n\}$ admits a
coarse embedding into Hilbert space. This extends an early result of J.
\v{S}pakula and R. Willett \cite{JR2013} to the case of metric spaces which may
not admit a coarse embedding into Hilbert space. Moreover, it implies that the
maximal coarse Baum-Connes conjecture holds for a large class of metric spaces
which may not admit a fibred coarse embedding into Hilbert space.
