A tautological continuous field of Roe bimodules

TL;DR

This paper extends the concept of continuous fields of C*-algebras to Hilbert C*-bimodules, constructing a tautological continuous field over a compactification of a partially ordered set related to coarse geometry.

Contribution

It introduces a new framework for continuous fields of Hilbert C*-bimodules based on TROs and applies it to coarse geometry via Roe bimodules and their associated topological structures.

Findings

Constructed a tautological continuous field of Hilbert C*-bimodules.

Connected the framework to coarse equivalence classes of metrics.

Provided a new perspective on Roe bimodules and their topological organization.

Abstract

We generalize the notion of a continuous field of C*-algebras to that of Hilbert C*-bimodules. Given a partially ordered set $P$ and a monotonically non-decreasing family of ternary rings of operators (TROs) assigned to the points of $P$, we equip $P$ with a certain zero-dimensional Hausdorff topology and use a certain compactification $\gamma P$ to get the base space for a continuous field of Hilbert C*-bimodules over $\gamma P$. As a motivating example, we consider the set $D(X,Y)$ of coarse equivalence classes of metrics on the disjoint union of two metric spaces, $X$ and $Y$. Each such class gives rise to a uniform Roe bimodule, a TRO linking the uniform Roe algebras of $X$ and $Y$. The resulting family of TROs is non-decreasing with respect to the natural partial order on $D(X,Y)$ and thus yields a tautological continuous field of Hilbert C*-bimodules over $\gamma D(X,Y)$.