The coarse Baum-Connes conjecture with filtered coefficients and product metric spaces

TL;DR

This paper introduces a generalized version of the coarse Baum-Connes conjecture with filtered coefficients, expanding its applicability to product metric spaces and providing new examples where the conjecture holds.

Contribution

It generalizes the coarse Baum-Connes conjecture using filtered coefficients, enabling the inclusion of product metric spaces and broadening the scope of the conjecture.

Findings

The conjecture with filtered coefficients is closed under products.

New examples of product metric spaces satisfying the conjecture are identified.

The routes to the original conjecture also apply to the generalized version.

Abstract

Inspired by the quantitative $K$-theory, in this paper, we introduce the coarse Baum-Connes conjecture with filtered coefficients which generalizes the original conjecture. There are two advantages for the conjecture with filtered coefficients. Firstly, the routes toward the coarse Baum-Connes conjecture also work for the conjecture with filtered coefficients. Secondly, the class of metric spaces that satisfy the conjecture with filtered coefficients is closed under products and yet it is unknown for the original conjecture. As an application, we discover some new examples of product metric spaces for the coarse Baum-Connes conjecture.