The author constructs a sequence of cubes in the infinitely dimensional hyperbolic space H∞ which is equi-coarsely equivalent to Z2n. As a corollary, it is proved that the infinitely dimensional hyperbolic space H∞ does not have property A.
Property A and uniform embeddability are notions of metric geometry which imply the coarse Baum-Connes conjecture and the Novikov conjecture.In this paper,the authors prove the permanence properties of property A and uniform embeddability of metric spaces under large scale decompositions of finite depth.
The notion of an ideal family of weighted subspaces of a discrete metric space X with bounded geometry is introduced. It is shown that, if X has Yu’s property A, the ideal structure of the Roe algebra of X with coefficients in B(H) is completely characterized by the ideal families of weighted subspaces of X, where B(H) denotes the C*-algebra of bounded linear operators on a separable Hilbert space H.
The author studies the metric spaces with operator norm localization property. It is proved that the operator norm localization property is coarsely invariant and is preserved under certain infinite union. In the case of finitely generated groups, the operator norm localization property is also preserved under the direct limits.
