The maximal coarse Baum–Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space

We introduce a notion of fibred coarse embedding into Hilbert space for metric spaces, which is a generalization of Gromovʼs notion of coarse embedding into Hilbert space. It turns out that a large class of expander graphs admit such an embedding. We show that the maximal coarse Baum–Connes conjecture holds for metric spaces with bounded geometry which admit a fibred coarse embedding into Hilbert space.