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 \(X\) .
