Property A On Discrete Metric Space

In 2000, G.Yu introduced a property on discrete metric spaces, which is called Prop-erty A. It is a weak form of amenability and has important applications to study the Novikov conjecture and group C*-algebra. Applying the method about uniform Roe algebra and positive definite kernel of Hilbert space, we show that: X is a discrete subspace with bounded geometry of Hilbert space H, if it satisfies (?) = 0 {N(R) = sup{#B(x,R): x∈X}), then X has Property A. We introduceα-quasi-geodesic metric space, which is a generalization of the quasi-geodesic metric space. And we show that Anα—quasi-geodesic metric space with bounded geometry has Property A, if its Hilbert space compression is greater than (?).