Property A and coarse embeddability for fuzzy metric spaces

TL;DR

This paper extends the concept of Property A to fuzzy metric spaces, demonstrating it as an invariant that ensures coarse embeddability into Hilbert space for these spaces.

Contribution

It introduces Property A for fuzzy metric spaces, proves its invariance under coarse equivalences, and characterizes it for uniformly locally finite fuzzy metric spaces.

Findings

Property A is an invariant in the coarse category of fuzzy metric spaces.

Fuzzy metric spaces with Property A are coarsely embeddable into Hilbert space.

Characterizations of Property A for uniformly locally finite fuzzy metric spaces are provided.

Abstract

Property A is a geometric property originally introduced for discrete metric spaces to provide a sufficient condition for coarse embeddability into Hilbert space, and it is defined via a F\o{}lner condition similar in spirit to the classical notion of amenability for groups. In this paper, we define property A for fuzzy metric spaces in the sense of George and Veeramani, show that it is an invariant in the coarse category of fuzzy metric spaces, and provide characterizations of it for uniformly locally finite fuzzy metric spaces. We also show that uniformly locally finite fuzzy metric spaces with property A are coarsely embeddable into Hilbert space.