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A tautological continuous field of Roe bimodules

Vladimir Manuilov

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 $γP$ to get the base space for a continuous field of Hilbert C*-bimodules over $γP$. 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 $γD(X,Y)$.

A tautological continuous field of Roe bimodules

Abstract

We generalize the notion of a continuous field of C*-algebras to that of Hilbert C*-bimodules. Given a partially ordered set and a monotonically non-decreasing family of ternary rings of operators (TROs) assigned to the points of , we equip with a certain zero-dimensional Hausdorff topology and use a certain compactification to get the base space for a continuous field of Hilbert C*-bimodules over . As a motivating example, we consider the set of coarse equivalence classes of metrics on the disjoint union of two metric spaces, and . Each such class gives rise to a uniform Roe bimodule, a TRO linking the uniform Roe algebras of and . The resulting family of TROs is non-decreasing with respect to the natural partial order on and thus yields a tautological continuous field of Hilbert C*-bimodules over .
Paper Structure (10 sections, 22 theorems, 12 equations)

This paper contains 10 sections, 22 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Continuous fields of Hilbert $C^*$-bimodules
  3. Continuous fields of Hilbert $C^*$-bimodules over posets
  4. A topology on posets and a compactification
  5. Ternary rings of operators
  6. Continuous field of $C^*$-bimodules
  7. Tautological continuous field of uniform Roe bimodules
  8. Poset of metrics on pairs of metric spaces
  9. Multiplicativity of $M_{[d]}$
  10. An example

Key Result

Proposition 2.5

Let $(T,M,\{M_t\}_{t\in T},\{\pi_t\}_{t\in T})$ be a continuous field of left Hilbert $C^*$-modules over a continuous field $(T,A,\{A_t\}_{t\in T},\{p_t\}_{t\in T})$ of $C^*$-algebras. Let $t_0\in T$, $\varepsilon>0$. Let $m\in M$ satisfy $p_{t_0}({}_A\langle m,m\rangle)=e\in A_{t_0}$ is a projectio

Theorems & Definitions (44)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • proof
  • ...and 34 more