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Twisted Roe algebras and their $K$-theory

Jintao Deng, Liang Guo

TL;DR

This work extends the coarse Baum-Connes framework by introducing twisted Roe algebras with coefficients and establishing a stability theorem for twisted CBC under coarse fibrations: if a fiber $Z$ and a base $Y$ both satisfy the twisted CBC with coefficients, then the total space $X$ of the coarse $Z$-fibration $Z\hookrightarrow X\to Y$ does as well. The authors develop a coarse imprimitivity approach, define twisted localization algebras, and employ a four-step proof to transfer CBC properties from base and fiber to the total space. They further show that the twisted CBC implies new results for group extensions and for spaces that coarsely embed into Hilbert space, culminating in a broad set of corollaries including CE-by-CE fibrations and product stability. Overall, the paper broadens the class of spaces and groups known to satisfy the coarse CBC with coefficients and provides robust tools for analyzing higher indices in coarse geometry. The results have potential impact on understanding Novikov-type conjectures and index theory on noncompact spaces and groups.

Abstract

In this paper, we introduce a notion of twisted Roe algebra and a twisted coarse Baum-Connes conjecture with coefficients. We will study the basic properties of twisted Roe algebras, including a coarse analogue of the imprimitivity theorem for metric spaces with a structure of coarse fibrations. We show that the twisted coarse Baum-Connes conjecture with coefficients holds for a metric space with a coarse fibration structure when the base space and the fiber satisfy the twisted coarse Baum-Connes conjecture with coefficients. As an application, the coarse Baum-Connes conjecture holds for a finitely generated group which is an extension of coarsely embeddable groups.

Twisted Roe algebras and their $K$-theory

TL;DR

This work extends the coarse Baum-Connes framework by introducing twisted Roe algebras with coefficients and establishing a stability theorem for twisted CBC under coarse fibrations: if a fiber and a base both satisfy the twisted CBC with coefficients, then the total space of the coarse -fibration does as well. The authors develop a coarse imprimitivity approach, define twisted localization algebras, and employ a four-step proof to transfer CBC properties from base and fiber to the total space. They further show that the twisted CBC implies new results for group extensions and for spaces that coarsely embed into Hilbert space, culminating in a broad set of corollaries including CE-by-CE fibrations and product stability. Overall, the paper broadens the class of spaces and groups known to satisfy the coarse CBC with coefficients and provides robust tools for analyzing higher indices in coarse geometry. The results have potential impact on understanding Novikov-type conjectures and index theory on noncompact spaces and groups.

Abstract

In this paper, we introduce a notion of twisted Roe algebra and a twisted coarse Baum-Connes conjecture with coefficients. We will study the basic properties of twisted Roe algebras, including a coarse analogue of the imprimitivity theorem for metric spaces with a structure of coarse fibrations. We show that the twisted coarse Baum-Connes conjecture with coefficients holds for a metric space with a coarse fibration structure when the base space and the fiber satisfy the twisted coarse Baum-Connes conjecture with coefficients. As an application, the coarse Baum-Connes conjecture holds for a finitely generated group which is an extension of coarsely embeddable groups.
Paper Structure (10 sections, 16 theorems, 111 equations)

This paper contains 10 sections, 16 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Twisted Roe algebra with coefficients
  3. Coarse algebras
  4. Roe algebras and twisted Roe algebras
  5. Coarse imprimitivity
  6. The coarse Baum-Connes conjecture and coarsely proper algebras
  7. The twisted coarse Baum-Connes conjecture
  8. The coarse Green-Julg's Theorem
  9. Proof of the main theorem
  10. Some Remarks

Key Result

Theorem 1.1

Let $Z\hookrightarrow X\to Y$ be a coarse $Z$-fibration metric space such that $X$, $Y$ and $Z$ have bounded geometry. For any coarse $Z$-algebra $\Gamma(F,\mathcal{A})$, we have the following Morita equivalence of two $C^*$-algebras: where $\Gamma_{ind}^Z(X, \mathcal{A})$ is the induced coarse $X$-algebra associated with the coarse fibration $Z \hookrightarrow X\to Y$ and the $Z$-algebra $\Gamma

Theorems & Definitions (46)

  • Theorem 1.1: Coarse imprimitivity theorem
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: Algebraic twisted Roe algebra
  • Example 2.6
  • ...and 36 more