Bounded geometry version of property A

TL;DR

This paper introduces a modified version of property A for metric spaces without bounded geometry, which still ensures coarse embeddability into Hilbert spaces and relates to uniform Roe algebras.

Contribution

It proposes a new property A variant for unbounded geometry spaces, extending coarse geometric analysis tools.

Findings

Modified property A implies coarse embeddability in Hilbert spaces.

Some non-property A unbounded spaces satisfy the new property A.

Establishes a relation between the new property A and uniform Roe algebras.

Abstract

For uniformly dicrete metric spaces without bounded geometry we suggest a modified version of property A based on metrics of bounded geometry greater than the given metric. We show that this version still implies coarse embeddability in Hilbert spaces, and that some examples of non-property A spaces of unbounded geometry satisfy this version. We also relate this version of property A to our version of uniform Roe algebras for spaces without bounded geometry and introduce an appropriate equivalence relation.