Metric sparsification and operator norm localization

TL;DR

This paper investigates the operator norm localization property in metric spaces, providing geometric conditions for it, and explores its implications for groups with infinite asymptotic dimension and sequences of expanding graphs.

Contribution

It introduces a new geometric condition for the operator norm localization property and applies it to groups with infinite asymptotic dimension.

Findings

Spaces with finite asymptotic dimension have the property.

Certain groups with infinite asymptotic dimension possess the property.

Sequences of expanding graphs do not have the property.

Abstract

We study an operator norm localization property and its applications to the coarse Novikov conjecture in operator K-theory. A metric space X is said to have operator norm localization property if there exists a positive number c such that for every r>0, there is R>0 for which, if m is a positive locally finite Borel measure on X, H is a separable infinite dimensional Hilbert space and T is a bounded linear operator acting on L^2(X,m) with propagation r, then there exists an unit vector v satisfying with support of diameter at most R and such that |Tv| is larger or equal than c|T|. If X has finite asymptotic dimension, then X has operator norm localization property. In this paper, we introduce a sufficient geometric condition for the operator norm localization property. This is used to give many examples of finitely generated groups with infinite asymptotic dimension and the operator norm localization property. We also show that any sequence of expanding graphs does not possess the operator norm localization property.