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

作者：Guo, Liang; Luo, Zheng; Wang, Qin; Zhang, Yazhou^*

来源：Journal of Topology and Analysis, 2023.

DOI：10.1142/S1793525323500073

• Coarse Novikov conjecture
• K-theory
• operator algebras
• higher index

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 ? : C-max(*)(X) ? C-*(X) from the maximal Roe algebra to the Roe algebra of X, and the canonical quotient map ? : C-u,max(*)(X) ? 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 ? N-n ? G(n) ? Q(n) ? 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 Spakula 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.