Generic quantum metric rigidity

TL;DR

This paper introduces a new algebraic framework for compact metric measure spaces and demonstrates that most such spaces lack nontrivial quantum symmetries, implying trivial classical automorphisms.

Contribution

It develops the concept of coherent algebra for metric measure spaces and shows that spaces with trivial quantum automorphism groups are generic in a topological sense.

Findings

Most compact metric measure spaces have trivial quantum automorphism groups.

Spaces with trivial quantum symmetries also have trivial classical automorphisms.

The coherent algebra concept bridges finite graph theory and metric measure space analysis.

Abstract

We introduce the coherent algebra of a compact metric measure space by analogy with the corresponding concept for a finite graph. As an application we show that upon topologizing the collection of isomorphism classes of compact metric measure spaces appropriately, the subset consisting of those with trivial compact quantum automorphism group is of second Baire category. The latter result can be paraphrased as saying that "most" compact metric measure spaces have no (quantum) symmetries; in particular, they also have trivial ordinary (i.e. classical) automorphism group.