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Quantum isometries and loose embeddings. (English) Zbl 1456.30098

Summary: 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.

MSC:

30L05 Geometric embeddings of metric spaces
46L85 Noncommutative topology
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