Ghostly ideals in uniform Roe algebras

TL;DR

This paper explores the ideal structure of uniform Roe algebras for general metric spaces, introducing ghostly ideals and analyzing their properties, with implications for K-theory and counterexamples to Baum-Connes conjectures.

Contribution

It introduces the concept of ghostly ideals in uniform Roe algebras, extending the understanding of their ideal lattice beyond property A spaces.

Findings

The geometric and ghostly ideals are the minimal and maximal elements in the ideal lattice.

Maximal ideals are characterized by minimal points in the Stone-ech boundary.

A criterion is provided for when geometric and ghostly ideals share the same K-theory.

Abstract

In this paper, we investigate the ideal structure of uniform Roe algebras for general metric spaces beyond the scope of Yu's property A. Inspired by the ideal of ghost operators coming from expander graphs and in contrast to the notion of geometric ideal, we introduce a notion of ghostly ideal in a uniform Roe algebra, whose elements are locally invisible in certain directions at infinity. We show that the geometric ideal and the ghostly ideal are respectively the smallest and the largest element in the lattice of ideals with a common invariant open subset of the unit space of the coarse groupoid by Skandalis-Tu-Yu, and hence the study of ideal structure can be reduced to classifying ideals between the geometric and the ghostly ones. As an application, we provide a concrete description for the maximal ideals in a uniform Roe algebra in terms of the minimal points in the Stone-\v{C}ech boundary of the space. We also provide a criterion to ensure that the geometric and the ghostly ideals have the same $K$-theory, which helps to recover counterexamples to the Baum-Connes type conjectures. Moreover, we introduce a notion of partial Property A for a metric space to characterise the situation in which the geometric ideal coincides with the ghostly ideal.