C*-Algebraic Characterization of Bounded Orbit Injection Equivalence for Minimal Free Cantor Systems

TL;DR

This paper characterizes bounded orbit injection equivalence for minimal free Cantor systems using C*-algebraic methods, introduces a new invariant, and explores conditions for strengthening this equivalence to orbit equivalence.

Contribution

It provides a C*-algebraic characterization of bounded orbit injection equivalence and introduces a new invariant that differs from the K_0 group, aiding in understanding orbit equivalences.

Findings

Bounded orbit injection equivalence characterized via Rieffel-Morita type of C*-algebras.

Constructed a new ordered group invariant for bounded orbit injection equivalence.

Identified conditions under which bounded orbit injection equivalence implies orbit or strong orbit equivalence.

Abstract

Bounded orbit injection equivalence is an equivalence relation defined on minimal free Cantor systems which is a candidate to generalize flip Kakutani equivalence to actions of the Abelian free groups on more than one generator. This paper characterizes bounded orbit injection equivalence in terms of a mild strengthening of Rieffel-Morita equivalence of the associated C*-crossed-product algebras. Moreover, we construct an ordered group which is an invariant for bounded orbit injection equivalence, and does not agrees with the K\_0 group of the associated C*-crossed-product in general. This new invariant allows us to find sufficient conditions to strengthen bounded orbit injection equivalence to orbit equivalence and strong orbit equivalence.