Fibring structures of ideals in Roe algebras and their $K$-theories

TL;DR

This paper explores the ideal structures within Roe algebras for metric spaces lacking Yu's property A, establishing fibring structures, defining ghostly and geometric ideals, and computing their $K$-theories to analyze obstructions to the coarse Baum-Connes conjecture.

Contribution

It introduces a novel fibring structure for ideals in Roe algebras, defines ghostly and geometric ideals, and calculates their $K$-theories, extending understanding beyond property A spaces.

Findings

Established fibring structures for ideals in Roe algebras.

Defined ghostly and geometric ideals and their borders.

Calculated $K$-theories of these ideals.

Abstract

In this paper, we investigate the ideal structure of Roe algebras for metric spaces beyond the scope of Yu's property A. Using the tool of rank distributions, we establish fibring structures for the lattice of ideals in Roe algebras and draw the border of each fibre by introducing the so-called ghostly ideals together with geometric ideals. We also provide coarse geometric criteria to ensure the coincidence of geometric and ghostly ideals and calculate their $K$-theories, which can be helpful to analyse obstructions to the coarse Baum-Connes conjecture on the level of ideals.