Open Cones and $K$-theory for $\ell^p$ Roe Algebras

TL;DR

This paper proves the $ ext{l}^p$ coarse Baum-Connes conjecture for open cones and demonstrates that the $K$-theory of $ ext{l}^p$ Roe algebras for these cones is independent of $p$, with applications to various geometric spaces.

Contribution

It verifies the $ ext{l}^p$ coarse Baum-Connes conjecture for open cones and shows $K$-theory independence of $p$, extending understanding in coarse geometry and operator algebras.

Findings

Verified the $ ext{l}^p$ coarse Baum-Connes conjecture for open cones.

Proved $K$-theory for $ ext{l}^p$ Roe algebras of open cones is independent of $p$.

Applied results to coarsely convex spaces including hyperbolic and CAT(0) spaces.

Abstract

In this paper, we verify the $\ell^p$ coarse Baum-Connes conjecture for open cones and show that the $K$-theory for $\ell^p$ Roe algebras of open cones are independent of $p\in[1,\infty)$. Combined with the result of T. Fukaya and S.-I. Oguni, we give an application to the class of coarsely convex spaces that includes geodesic Gromov hyperbolic spaces, CAT(0)-spaces, certain Artin groups and Helly groups equipped with the word length metric.