Relating the Roe Algebra of a Space to the Uniform Roe Algebras of Its Discretizations

Abstract

The Roe algebra \(C^{*}(X)\) is a non-commutative \(C^{*}\)-algebra reflecting metric properties of a space \(X\), and it is interesting to understand relation between the Roe algebra of \(X\) and the uniform Roe algebra of its discretizations. Here we construct, for a simplicial space \(X\), a continuous field of \(C^{*}\)-algebras over \(\mathbb{N}\cup\{\infty\}\) with the fibers over finite points the uniform \(C^{*}\)-algebras of discretizations of \(X\), and the fiber over \(\infty\) the Roe algebra of \(X\). We also construct the direct limit of the uniform Roe algebras of discretizations and its embedding into the Roe algebra of \(X\).