Constructing minimal homeomorphisms on point-like spaces and a dynamical presentation of the Jiang-Su algebra. (English) Zbl 1416.46057
The Jiang-Su algebra \(\mathcal{Z}\) (which was first introduced by X.-H. Jiang and H.-B. Su [Am. J. Math. 121, No. 2, 359–413 (1999; Zbl 0923.46069)]) is the unique simple infinite-dimensional \(C^*\)-algebra which is an inductive limit of finite direct sums of dimension drop algebras with its Elliott invariant being the same as that of the complex field \(\mathbb{C}\). It plays an important role in the study of the Elliott conjecture.
In parallel with measure dynamical presentations of the hyperfinite \(\mathrm{II}_1\)-factor, the authors show that \(\mathcal{Z}\) is isomorphic to the \(C^*\)-algebra of a minimal étale equivalence relation.
More precisely, suppose that \(\varphi\) is a minimal diffeomorphism on the \(d\)-dimensional sphere \(S^d\) with \(d \geq 3\) being an odd integer. Through a limiting process, the authors construct an infinite-dimensional connected compact metric space \(Z_\varphi\) with its covering dimension being dominated by \(d\), whose cohomology and \(K\)-theory are the same as that of a point, a minimal homeomorphism \(\xi_\varphi\) in \(Z_\varphi\), as well as a continuous surjection \(q:Z_\varphi\to S^d\), respecting \(\varphi\) and \(\xi_\varphi\) such that \(q\) induces an affine bijection between the set of \(\xi_\varphi\)-invariant Borel probability measures on \(Z_\varphi\) and that of \(\varphi\)-invariant Borel probability measures on \(S^d\). The orbit eqivalence relation of \(\xi_\varphi\) is denoted by \(\mathcal{E}_{\xi_\varphi}\), and it admits a natural topology that turns it into an étale groupoid. Fix a point \(z\in Z_\varphi\). The authors define the orbit-breaking equivalence relation \(\mathcal{E}_z\), which is an open subset of \(\mathcal{E}_{\xi_\varphi}\) (hence it is also étale) and is a minimal equivalence relation.
If \(\varphi\) is a uniquely ergodic miminal diffeomorphism (whose existence was established in [A. Fathi and M. R. Herman, Astérisque 49, 37–59 (1978; Zbl 0374.58010)]), the authors show that the groupoid \(C^*\)-algebra \(C^*(\mathcal{E}_z)\) admits the same Elliott invariant as \(\mathcal{Z}\), and that \(C^*(\mathcal{E}_z)\otimes \mathcal{Z} \cong C^*(\mathcal{E}_z)\). They then employ the classification results in [H.-X. Lin and Z. Niu, Adv. Math. 219, No. 5, 1729–1769 (2008; Zbl 1162.46033); J. Funct. Anal. 260, No. 1, 1–29 (2011; Zbl 1213.46057)] to show that \(C^*(\mathcal{E}_z)\cong \mathcal{Z}\).
Furthermore, in the case when \(\varphi\) is not necessarily uniquely ergodic, the authors give a classification result for algebras of the form \(C(Z_\varphi)\rtimes \mathbb{Z}\) (and another one for algebras of the form \(C^*(\mathcal{E}_z)\)), in terms of their tracial state spaces.
In parallel with measure dynamical presentations of the hyperfinite \(\mathrm{II}_1\)-factor, the authors show that \(\mathcal{Z}\) is isomorphic to the \(C^*\)-algebra of a minimal étale equivalence relation.
More precisely, suppose that \(\varphi\) is a minimal diffeomorphism on the \(d\)-dimensional sphere \(S^d\) with \(d \geq 3\) being an odd integer. Through a limiting process, the authors construct an infinite-dimensional connected compact metric space \(Z_\varphi\) with its covering dimension being dominated by \(d\), whose cohomology and \(K\)-theory are the same as that of a point, a minimal homeomorphism \(\xi_\varphi\) in \(Z_\varphi\), as well as a continuous surjection \(q:Z_\varphi\to S^d\), respecting \(\varphi\) and \(\xi_\varphi\) such that \(q\) induces an affine bijection between the set of \(\xi_\varphi\)-invariant Borel probability measures on \(Z_\varphi\) and that of \(\varphi\)-invariant Borel probability measures on \(S^d\). The orbit eqivalence relation of \(\xi_\varphi\) is denoted by \(\mathcal{E}_{\xi_\varphi}\), and it admits a natural topology that turns it into an étale groupoid. Fix a point \(z\in Z_\varphi\). The authors define the orbit-breaking equivalence relation \(\mathcal{E}_z\), which is an open subset of \(\mathcal{E}_{\xi_\varphi}\) (hence it is also étale) and is a minimal equivalence relation.
If \(\varphi\) is a uniquely ergodic miminal diffeomorphism (whose existence was established in [A. Fathi and M. R. Herman, Astérisque 49, 37–59 (1978; Zbl 0374.58010)]), the authors show that the groupoid \(C^*\)-algebra \(C^*(\mathcal{E}_z)\) admits the same Elliott invariant as \(\mathcal{Z}\), and that \(C^*(\mathcal{E}_z)\otimes \mathcal{Z} \cong C^*(\mathcal{E}_z)\). They then employ the classification results in [H.-X. Lin and Z. Niu, Adv. Math. 219, No. 5, 1729–1769 (2008; Zbl 1162.46033); J. Funct. Anal. 260, No. 1, 1–29 (2011; Zbl 1213.46057)] to show that \(C^*(\mathcal{E}_z)\cong \mathcal{Z}\).
Furthermore, in the case when \(\varphi\) is not necessarily uniquely ergodic, the authors give a classification result for algebras of the form \(C(Z_\varphi)\rtimes \mathbb{Z}\) (and another one for algebras of the form \(C^*(\mathcal{E}_z)\)), in terms of their tracial state spaces.
Reviewer: Chi-Keung Ng (Tianjin)
MSC:
| 46L05 | General theory of \(C^*\)-algebras |
| 46L35 | Classifications of \(C^*\)-algebras |
| 37A55 | Dynamical systems and the theory of \(C^*\)-algebras |
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