Relating Roe algebras to uniform Roe algebras of discretizations

Roe algebras play an increasingly important role in the index theory of elliptic operators on noncompact manifolds and their generalizations. Following the ideology of non-commutative geometry, they provide an interaction between metric spaces (e.g. manifolds) and (non-commutative) $C^*$-algebras.
If the metric space is discrete, along with the Roe algebra, one can define a uniform Roe algebra, which is quite different from the Roe algebra. It is easier to compute with, but has fewer connections with elliptic theory.
Manifolds and some other spaces $X$ are often endowed with discrete subspaces $D\subset X$ which are $\varepsilon$-dense for some $\varepsilon$, such as lattices in Lie groups or, more generally, Delone sets in metric spaces. Some problems related to $X$ can be simplified by reducing to such discretizations $D$. In particular, it would be interesting to understand the connection between the Roe algebra of a space $X$ and the uniform Roe algebras of its discretizations. We restrict ourselves to the case where $X$ is a simplicial complex with the standard metric and the set of its vertices is a discretization of $X$, and to obtain denser discretizations, we will partition the simplices into smaller equal simplices.
Our first result is a construction of a continuous field of $C^*$-algebras over $\mathbb N\cup\{\infty\}$ such that the fiber over $\infty$ is the Roe algebra of $X$, and the fiber over any finite point $n$ is the uniform Roe algebra of the corresponding discretization $D(n)$, $n\in\mathbb N$, of the space $X$. Such non locally trivial continuous fields of $C^*$-algebras are interesting because they provide a connection between fibers at different points. In particular, they can provide a map from the $K$-theory of the fiber over $\infty$ to the $K$-theory of fibers over finite points.
The second result is the construction of a direct limit of uniform Roe algebras of discretizations $D(n)$ of space $X$ that are increasingly dense, and embedding this direct limit in the Roe algebra of $X$. Such direct limit is a good candidate for what could be the uniform Roe algebra of a non-discrete space.