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A tautological continuous field of Roe bimodules
Vladimir Manuilov
https://arxiv.org/abs/2603.23366 https://arxiv.org/pdf/2603.23366 https://arxiv.org/html/2603.23366
arXiv:2603.23366v1 Announce Type: new
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 $\gamma D(X,Y)$.
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