Expanders are counterexamples to the \(\ell^p\) coarse Baum-Connes conjecture (Q6039646)

The main result of this article shows that certain counterexamples for the usual coarse Baum-Connes conjecture remain counterexamples for the \(\ell^p\)-version of the conjecture. More precisely, the \(\ell^p\)-version of the coarse Baum-Connes assembly map fails to be surjective for certain box spaces and for \(1<p<\infty\). The box space must be built from a residually finite hyperbolic group and be an expander. This implies a version of property T, which allows to build an \(\ell^p\)-version of the Kazhdan projection. This projection is a a ghost idempotent, but not compact. Such idempotents do not belong to the image of the \(\ell^p\)-version of the coarse Baum-Connes assembly map. Along the way, the article also extends relevant tools to the \(\ell^p\)-setting, such as the operator norm localization property and ghost projections. In an appendix, it is shown that some potentially different versions of the coarse Baum-Connes assembly have equivalent \(\ell^p\)-generalisations.