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On the Baum-Connes conjecture for $D_{\infty}$

Eugenia Ellis, Emanuel Rodríguez Cirone, Gisela Tartaglia

TL;DR

The paper provides an explicit Higson–Kasparov–style proof of the Baum–Connes conjecture with coefficients for the infinite dihedral group $D_ obreak o ext{D}_ obreak ty$. By constructing a proper $D_ obreak o$-algebra $\\mathcal{C}(\\mathbb{R})$ from the Clifford algebra and producing $E_{D_ obreak o}$-theory elements $\\beta$ and $\\alpha$ with $\\alpha\\circ\\beta=1$, the authors deduce that the assembly map is an isomorphism for all coefficients. The argument specializes the HK framework to a finite-dimensional affine action, yielding a concrete computation of $K_0(C_r^*(D_ obreak o))\\cong \\mathbb{Z}^3$ and $K_1(C_r^*(D_ obreak o))=0$, and aligns with the Davis–Lück formulation. This work adds a hands-on example to the class of groups for which the Baum–Connes conjecture is verified, illustrating the practical use of asymptotic morphisms and graded $G$-$C^*$-algebras in index theory.

Abstract

We make an exposition of the proof of the Baum-Connes conjecture for the infinite dihedral group following the ideas of Higson and Kasparov.

On the Baum-Connes conjecture for $D_{\infty}$

TL;DR

The paper provides an explicit Higson–Kasparov–style proof of the Baum–Connes conjecture with coefficients for the infinite dihedral group . By constructing a proper -algebra from the Clifford algebra and producing -theory elements and with , the authors deduce that the assembly map is an isomorphism for all coefficients. The argument specializes the HK framework to a finite-dimensional affine action, yielding a concrete computation of and , and aligns with the Davis–Lück formulation. This work adds a hands-on example to the class of groups for which the Baum–Connes conjecture is verified, illustrating the practical use of asymptotic morphisms and graded --algebras in index theory.

Abstract

We make an exposition of the proof of the Baum-Connes conjecture for the infinite dihedral group following the ideas of Higson and Kasparov.

Paper Structure

This paper contains 11 sections, 6 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graded G-C*-algebras
  4. Maximal tensor product
  5. Crossed products
  6. Asymptotic morphisms
  7. $E$-Theory groups
  8. The Baum-Connes assembly map
  9. The Baum-Connes conjecture for the infinite dihedral group
  10. The element beta
  11. The element alpha

Key Result

Theorem 2.5

libro-higson-connes*Thm. 2.20 Let $G$ be a countable discrete group. Suppose there exists a proper $G$-$C^*$-algebra $B$ and elements $\beta \in E_G(\mathbb{C},B)$ and $\alpha \in E_G(B,\mathbb{C})$ such that Then the Baum-Connes assembly map $\mu: K^{top}(G,D)\to K(C^*(G,D))$ is an isomorphism for every separable $G$-$C^*$-algebra $D$.

Theorems & Definitions (14)

  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 4 more