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

Abstract

Let [Formula: see text] be a discrete metric space with bounded geometry. In this paper, we show that if [Formula: see text] admits an “A-by-CE” coarse fibration, then the canonical quotient map [Formula: see text] from the maximal Roe algebra to the Roe algebra of [Formula: see text], and the canonical quotient map [Formula: see text] from the maximal uniform Roe algebra to the uniform Roe algebra of [Formula: see text], induce isomorphisms on [Formula: see text]-theory. A typical example of such a space arises from a sequence of group extensions [Formula: see text] such that the sequence [Formula: see text] has Yu’s property A, and the sequence [Formula: see text] 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.