Compact Metrizable Structures and Classification Problems

Abstract

We introduce and study the framework of compact metric structures and their associated notions of isomorphisms such as homeomorphic and bi-Lipschitz isomorphism. This is subsequently applied to model various classification problems in analysis such as isomorphism of C*-algebras and affine homeomorphism of Choquet simplices, where among other things we provide a simple proof of the completeness of the isomorphism relation of separable, simple, nuclear C*-algebras recently established by M. Sabok.

Classification of One Dimensional Dynamical Systems by Countable Structures

“…Consider structures of the form (Q, R) where R is a closed binary relation on Q. Two such structures (Q, R) and (Q, S) are said to be isomorphic if there is a homeomorphism : Q → Q for which ( × )(R) = S. By a fairly more general result [24] it follows that such isomorphism equivalence relation is Borel reducible to E G∞ . There is a Borel (even continuous) reduction which takes a continuous map f : Q → Q and assigns (Q, graph(f)) to it.…”

Classification of one dimensional dynamical systems by countable structures

“…Consider structures of the form (Q, R) where R is a closed binary relation on Q. Two such structures (Q, R) and (Q, S) are said to be isomorphic if there is a homeomorphisms ψ : Q → Q for which (ψ × ψ)(R) = S. By a fairly more general result [RZ18] it follows that such isomorphism equivalence relation is Borel reducible to E G∞ . There is a Borel (even continuous) reduction which takes a continuous map f : Q → Q and assigns (Q, graph(f )) to it.…”