Quantum isometries and loose embeddings

Abstract

We show that countable metric spaces always have quantum isometry groups, thus extending the class of metric spaces known to possess such universal quantum-group actions. Motivated by this existence problem we define and study the notion of loose embeddability of a metric space $(X,d_X)$ into another, $(Y,d_Y)$: the existence of an injective continuous map that preserves both equalities and inequalities of distances. We show that $0$-dimensional compact metric spaces are "generically" loosely embeddable into the real line, even though not even all countable metric spaces are.

9 pages + references; material being split off at the referee's recommendation

References in corpus (1)

• On quantum symmetries of compact metric spaces

Cited by in corpus (1)

• (In)equality distance patterns and embeddability into Hilbert spaces