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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 $X$, induceisomorphisms on $K$-theory. A typical example of such a space arises from asequence of group extensions $\{1\to N_n\to G_n\to Q_n\to 1\}$ such that thesequence $\{N_n\}$ has Yu's property A, and the sequence $\{Q_n\}$ admits acoarse 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 maynot admit a coarse embedding into Hilbert space. Moreover, it implies that themaximal coarse Baum-Connes conjecture holds for a large class of metric spaceswhich may not admit a fibred coarse embedding into Hilbert space.

K-theory of the maximal and reduced Roe algebras of metric spaces with A-by-CE coarse fibrations