Hilbert C*-modules related to discrete metric spaces

TL;DR

This paper explores how the metric on the union of two sets induces a Hilbert C*-module over the uniform Roe algebra of one set, providing examples and analyzing their properties.

Contribution

It introduces a novel construction of Hilbert C*-modules from metrics on unions of spaces, linking metric geometry with operator algebras.

Findings

The metric on the union defines a Hilbert C*-module over the uniform Roe algebra.

Several explicit examples of such Hilbert C*-modules are provided.

The paper establishes connections between metric properties and operator algebra structures.

Abstract

It is shown that the metric on the union of the sets $X$ and $Y$ defines a Hilbert $C^*$-module over the uniform Roe algebra of the space $X$ with a fixed metric $d_X$. A number of examples of such Hilbert $C^*$-modules are described.