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: Cmax∗(X) → C∗(X) from the maximal Roe algebra to the Roe algebra of X, and the canonical quotient map λ: Cu,max∗(X) → Cu∗(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 → 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 Špakula and Willett [Maximal and reduced Roe algebras of coarsely embeddable spaces, J. Reine Angew. Math. 678 (2013) 35-68] 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 fibered coarse embedding into Hilbert space.