Abstract:
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.
