Quantum metrics on noncommutative spaces

TL;DR

This paper proposes two new definitions of quantum metrics on noncommutative spaces using tensor products, exploring their properties and connections to existing theories, but lacks concrete nonclassical examples.

Contribution

It introduces novel formulations of quantum metrics on noncommutative spaces and examines their fundamental properties and links to Rieffel's theory.

Findings

Defined quantum metrics as elements of the spatial tensor product

Explored basic properties and theoretical connections

Identified the absence of nonclassical examples in matrix C*-algebras

Abstract

We introduce two new formulations for the notion of "quantum metric on noncommutative space". For a compact noncommutative space associated to a unital C*-algebra, our quantum metrics are elements of the spatial tensor product of the C*-algebra with itself. We consider some basic properties of these new objects, and state some connections with the Rieffel theory of compact quantum metric spaces. The main gap in our work is the lack of a nonclassical example even in the case of C*-algebras of matrices.