$\ell^p$-coarse Baum-Connes conjecture for $\ell^{q}$-coarse embeddable spaces

TL;DR

This work establishes an $oldsymbol{\, iem_p}$-version of the coarse Baum–Connes conjecture for spaces with bounded geometry that coarsely embed into $oldsymbol{\, iem_q}$, for all $p,q\in[1,\infty)$. The authors build an $oldsymbol{\, iem_p}$-Roe algebra framework and a Bott–Dirac operator constructed via the Mazur map between $oldsymbol{\, iem_q}$ and $oldsymbol{\, iem_2}$, twisting $K$-theory classes and controlling the resulting mixed $oldsymbol{\, iem_p}$–$oldsymbol{\, iem_2}$ norms with a vector-valued Marcinkiewicz–Zygmund inequality. A twisted Roe algebra setup and a Mayer–Vietoris descent argument reduce the problem to finite-dimensional local data, yielding the isomorphism of the $oldsymbol{\, iem_p}$-assembly map and, consequently, a new path to Novikov-type results via Roe descent. As a corollary, the $K$-theory of the $oldsymbol{\, iem_p}$-Roe algebra is independent of $p$ under the stated coarse embeddability assumptions.

Abstract

We prove an $\ell^p$-version of the coarse Baum-Connes conjecture for spaces that coarsely embedds into $\ell^q$-spaces for any $p$ and $q$ in $[1,\infty)$.

Theorems & Definitions (58)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Definition 2.4
• Conjecture 2.5: Coarse Baum--Connes Conjecture
• Lemma 2.6
• proof
• Lemma 2.7