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On the quantitative coarse Baum-Connes conjecture with coefficients

Jianguo Zhang

Abstract

In this paper, we introduce the quantitative coarse Baum-Connes conjecture with coefficients (or QCBC, for short) for proper metric spaces which refines the coarse Baum-Connes conjecture. And we prove that QCBC is derived by the coarse Baum-Connes conjecture with coefficients which provides many examples satisfying QCBC. In the end, we show QCBC can be reduced to the uniformly quantitative coarse Baum-Connes conjecture with coefficients of a sequence of bounded metric spaces.

On the quantitative coarse Baum-Connes conjecture with coefficients

Abstract

In this paper, we introduce the quantitative coarse Baum-Connes conjecture with coefficients (or QCBC, for short) for proper metric spaces which refines the coarse Baum-Connes conjecture. And we prove that QCBC is derived by the coarse Baum-Connes conjecture with coefficients which provides many examples satisfying QCBC. In the end, we show QCBC can be reduced to the uniformly quantitative coarse Baum-Connes conjecture with coefficients of a sequence of bounded metric spaces.

Paper Structure

This paper contains 12 sections, 44 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. A revisit to the quantitative $K$-theory
  3. Basic concepts
  4. Properties of the quantitative $K$-theory
  5. the quantitative coarse Baum-Connes conjecture
  6. Roe algebras with coefficients
  7. Localization algebras with coefficients
  8. the quantitative coarse Baum-Connes conjecture with coefficients
  9. Relations between the quantitative coarse Baum-Connes conjecture with coefficients and the original conjecture
  10. $K$-theory for localization algebras with coefficients for a sequence of metric spaces
  11. Relations
  12. Reducing the conjecture to a sequence of bounded metric spaces

Key Result

Theorem 1.1

Let $A$ be a $C^{\ast}$-algebra, the quantitative coarse Baum-Connes conjecture with coefficients in $A$ implies the coarse Baum-Connes conjecture with coefficients in $A$.

Theorems & Definitions (107)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 97 more