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

Let X be a discrete metric space with bounded geometry. In this paper, we show that if X admits an "A-by-CE coarse fibration", then the canonical quotient map λ : C max(X) → C (X) from the maximal Roe algebra to the Roe algebra of X, and the canonical quotient map λ : C u,max(X) → C ∗ u(X) from the maximal uniform Roe algebra to the uniform Roe algebra of X, induce isomorphisms on Ktheory. A typical example of such a space arises from a sequence of group extensions {1 → Nn → Gn → Qn → 1} such that the sequence {Nn} has Yu’s property A, and the sequence {Qn} admits a coarse embedding into Hilbert space. This extends an early result of J. Špakula and R. Willett [24] 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.