Expanders are counterexamples to the $\ell^p$ coarse Baum-Connes conjecture

Abstract

We show that certain expanders are counterexamples to the coarse p-Baum-Connes conjecture. Contents 1 Introduction 1 2 Preliminaries 4 3 Lifting maps, p-operator norm localization, and surjectivity of the assembly map 8 3.1 The map φ L .

$L^p$ coarse Baum–Connes conjecture and $K$-theory for $L^p$ Roe algebras

“…By Propositions 5.18 and 5.19, we obtain that the K-theory for L p localization algebra is independent of p for a finite dimensional simplicial complex. This gives a partial answer to Question 26 in [9] proposed by Chung and Nowak. Proposition 5.20.…”

“…In [18,41], Higson-Lafforgue-Skandalis and Willett-Yu showed that Magulis-type expanders and expanders with large girth are counter-examples for the surjectivity of the coarse Baum-Connes conjecture. In [9], Chung and Nowak showed that Margulistype expanders are still a counterexample for the L p coarse Baum-Connes conjecture. However, the existence of an injectivity counterexample of the L p coarse Baum-Connes conjecture is still open.…”