# Banach Algebras Associated to Metric Operator Fields

## TL;DR

This paper introduces the concept of metric operator fields in noncommutative geometry, establishing their connection to Banach *-algebras, and explores their properties, examples, and potential applications to quantum gravity.

## Contribution

It defines metric operator fields and constructs associated Lipschitz algebras, linking noncommutative geometry with operator algebra theory and quantum physics.

## Key findings

- Associated Lipschitz algebras are Banach *-algebras.
- For von Neumann algebra-valued fields, Lipschitz algebras are dual Banach spaces.
- Under certain conditions, these algebras are non-amenable.

## Abstract

Motivated by noncommutative geometry and quantum physics, the concept of `metric operator field' is introduced. Roughly speaking, a metric operator field is a vector field on a set with values in self tensor product of a bundle of C*-algebras, satisfying properties similar to an ordinary metric (distance function). It is proved that to any such object there naturally correspond a Banach *-algebra that we call Lipschitz algebra, a class of probabilistic metrics, and (under some conditions) a (nontrivial) continuous field of C*-algebras in the sense of Dixmier. It is proved that for metric operator fields with values in von Neumann algebras the associated Lipschitz algebras are dual Banach spaces, and under some conditions, they are not amenable Banach algebras. Some examples and constructions are considered. We also discuss very briefly a possible application to quantum gravity.