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**Topological orbit equivalence and \(C^*\)-crossed products.**
*(English)*
Zbl 0834.46053

Let \((X, F)\) be a Cantor minimal system, i.e. \(F\) is a homeomorphism of the Cantor set \(X\) with all \(F\)-orbits being dense. The \(C^*\)-crossed product associated to \((X, F)\) is a simple \(C^*\)-algebra of real rank zero, which is an inductive limit of circle algebras and with \(K_1\)- group equal to \(\mathbb{Z}\). The ordered \(K_0\)-group of the crossed product, which by G. Elliott’s theorem is a complete isomorphism invariant, is a simple dimension group, and is order isomorphic (preserving the canonical order units) to the ordered group \(K^0(X, F)\) of \((X, F)\), the latter group being defined in purely dynamical terms.

In the paper, it is proved that \(K^0(X, F)\) as an ordered group with canonical order unit is a complete invariant for strong orbit equivalence. As a consequence one gets a theorem in the topological/\(C^*\)-algebra setting which is analogous to Krieger’s theorem in the measure-theoretic ergodic/von Neumann algebra setting.

Two Cantor minimal systems \((X_1, F_1)\) and \((X_2, F_2)\) are (topological) orbit equivalent if there exists a homeomorphism \(T: X_1\to X_2\) mapping the \(F_1\)-orbit of \(x\) in \(X_1\) onto the \(F_2\)-orbit of \(T(x)\), for all \(x\) in \(X_1\). If there exists such an orbit map \(T\) so that the two associated orbit cocycles each have at most one point of discontinuity, we say that \((X_1, F_1)\) and \((X_2, F_2)\) are strong orbit equivalent. The concept of strong orbit equivalence is put into perspective by Boyle’s theorem, which says that if the orbit cocycles are continuous everywhere, then the two systems are (flip) conjugate.

The main theorem of the paper states that a complete invariant for the orbit equivalence class of the Cantor minimal system \((X, F)\) is \(K^0(X, F)/\text{Inf}(K^0(X, F))\) as an ordered group with canonical under unit, where \(\text{Inf}(K^0(X, F))\) denotes the infinitesimal subgroup of \(K^0(X, F)\). (The theorem is extended to also comprise AF- systems.) A corollary of the theorem is a result with a Dye flavour to it (in the measure-preserving ergodic case), namely: Every uniquely ergodic Cantor minimal system is either orbit equivalent to an odometer system, or to a Denjoy system (i.e. an aperiodic homeomorphism of the unit circle which is not conjugate to a pure rotation – restricted to the support of its unique invariant probability measure, this set being a Cantor set).

The proof of the main theorem requires the application of techniques and results from a wide variety of sources. There are two key ingredients to the proof: One is the application of a model theorem for Cantor minimal systems based upon ordered Bratteli diagrams; the other is a homological algebra result to the effect that there is a dynamical realization of every abelian extension of the free abelian group \(\bigoplus^\infty_1 \mathbb{Z}\) by a given dimension group \(G\).

The paper also treats the case if one has order isomorphism of \(K^0\)- groups, dropping the condition of preserving order units, and shows this is related to Kakutani orbit equivalence.

Finally, a theorem is proved clarifying what happens if the cocycles associated to orbit maps have a finite set of discontinuity points.

In the paper, it is proved that \(K^0(X, F)\) as an ordered group with canonical order unit is a complete invariant for strong orbit equivalence. As a consequence one gets a theorem in the topological/\(C^*\)-algebra setting which is analogous to Krieger’s theorem in the measure-theoretic ergodic/von Neumann algebra setting.

Two Cantor minimal systems \((X_1, F_1)\) and \((X_2, F_2)\) are (topological) orbit equivalent if there exists a homeomorphism \(T: X_1\to X_2\) mapping the \(F_1\)-orbit of \(x\) in \(X_1\) onto the \(F_2\)-orbit of \(T(x)\), for all \(x\) in \(X_1\). If there exists such an orbit map \(T\) so that the two associated orbit cocycles each have at most one point of discontinuity, we say that \((X_1, F_1)\) and \((X_2, F_2)\) are strong orbit equivalent. The concept of strong orbit equivalence is put into perspective by Boyle’s theorem, which says that if the orbit cocycles are continuous everywhere, then the two systems are (flip) conjugate.

The main theorem of the paper states that a complete invariant for the orbit equivalence class of the Cantor minimal system \((X, F)\) is \(K^0(X, F)/\text{Inf}(K^0(X, F))\) as an ordered group with canonical under unit, where \(\text{Inf}(K^0(X, F))\) denotes the infinitesimal subgroup of \(K^0(X, F)\). (The theorem is extended to also comprise AF- systems.) A corollary of the theorem is a result with a Dye flavour to it (in the measure-preserving ergodic case), namely: Every uniquely ergodic Cantor minimal system is either orbit equivalent to an odometer system, or to a Denjoy system (i.e. an aperiodic homeomorphism of the unit circle which is not conjugate to a pure rotation – restricted to the support of its unique invariant probability measure, this set being a Cantor set).

The proof of the main theorem requires the application of techniques and results from a wide variety of sources. There are two key ingredients to the proof: One is the application of a model theorem for Cantor minimal systems based upon ordered Bratteli diagrams; the other is a homological algebra result to the effect that there is a dynamical realization of every abelian extension of the free abelian group \(\bigoplus^\infty_1 \mathbb{Z}\) by a given dimension group \(G\).

The paper also treats the case if one has order isomorphism of \(K^0\)- groups, dropping the condition of preserving order units, and shows this is related to Kakutani orbit equivalence.

Finally, a theorem is proved clarifying what happens if the cocycles associated to orbit maps have a finite set of discontinuity points.

Reviewer: T.Giordano (Ottawa)

### MSC:

46L80 | \(K\)-theory and operator algebras (including cyclic theory) |

46L35 | Classifications of \(C^*\)-algebras |

46L55 | Noncommutative dynamical systems |

54H20 | Topological dynamics (MSC2010) |