Category: Coarse Geometry

This is a continuation of the previous two posts I and II. In the first one I defined the uniform Roe algebra \(C_u^*(X)\) of a metric space \(X\) of bounded geometry and mentioned the following interesting result: Two metric spaces \(X\) and \(Y\) are coarsely equivalent if and only if their uniform Roe algebras \(C_u^*(X)\) … Continue reading "Metric spaces and C*-algebras III"

In the previous post about metric spaces and C*-algebras we saw the definition of the uniform Roe algebra \(C^*_u(X)\) of a metric space \(X\) and discussed its remarkable property that two metric spaces (of bounded geometry) \(X\) and \(Y\) are coarsely equivalent if and only if their uniform Roe algebras \(C^*_u(X)\) and \(C^*_u(Y)\) are Morita … Continue reading "Metric spaces and C*-algebras II"

Let \(X\) be a metric space of bounded geometry. The latter means that for all \(r > 0\) exists an \(n_r \in \mathbb{N}\) such that every ball in \(X\) of radius \(r\) has at most \(n_r\) elements. This especially implies that \(X\) is discrete. It might seem a bit strange for some to consider discrete … Continue reading "Metric spaces and C*-algebras"

Let \(X\) and \(E\) be Banach spaces. The metrics on these spaces induce both uniform and coarse structures, and we can ask whether the following two statements are equivalent to each other: \(X\) uniformly embeds into \(E\). \(X\) coarsely embeds into \(E\). Let us first write down what it means for a map \(\phi \colon … Continue reading "Uniform and coarse embeddings of Banach spaces"