TY - JOUR

T1 - Strongly outer actions of amenable groups on Z-stable nuclear C*-algebras

AU - Gardella, Eusebio

AU - Hirshberg, Ilan

AU - Vaccaro, Andrea

N1 - Funding Information:
The first named author was partially supported by the Deutsche Forschungsgemeinschaft through an eigene Stelle and under Germany's Excellence Strategy EXC 2044 - 390685587 (Mathematics Münster: Dynamics-Geometry-Structure), and by a Postdoctoral Research Fellowship from the Humboldt Foundation . This work was supported by German-Israeli Foundation grant 1137/2011 , by Israel Science Foundation Grant 476/16 and by the Fields Institute . The third author was supported by the European Union's European Research Council Marie Sklodowska-Curie grant No. 891709 .
Publisher Copyright:
© 2022 The Author(s)

PY - 2022/6/1

Y1 - 2022/6/1

N2 - Let A be a separable, unital, simple, Z-stable, nuclear C⁎-algebra, and let α:G→Aut(A) be an action of a discrete, countable, amenable group. Suppose that the orbits of the action of G on T(A) are finite and that their cardinality is bounded. We show that the following are equivalent: (1) α is strongly outer; (2) α⊗idZ has the weak tracial Rokhlin property. If G is moreover residually finite, the above conditions are also equivalent to (3) α⊗idZ has finite Rokhlin dimension (in fact, at most 2). If ∂eT(A) is furthermore compact, has finite covering dimension, and the orbit space ∂eT(A)/G is Hausdorff, we generalize results by Matui and Sato to show that α is cocycle conjugate to α⊗idZ, even if α is not strongly outer. In particular, in this case the equivalences above hold for α in place of α⊗idZ. In the course of the proof, we develop equivariant versions of complemented partitions of unity and uniform property Γ as technical tools of independent interest.

AB - Let A be a separable, unital, simple, Z-stable, nuclear C⁎-algebra, and let α:G→Aut(A) be an action of a discrete, countable, amenable group. Suppose that the orbits of the action of G on T(A) are finite and that their cardinality is bounded. We show that the following are equivalent: (1) α is strongly outer; (2) α⊗idZ has the weak tracial Rokhlin property. If G is moreover residually finite, the above conditions are also equivalent to (3) α⊗idZ has finite Rokhlin dimension (in fact, at most 2). If ∂eT(A) is furthermore compact, has finite covering dimension, and the orbit space ∂eT(A)/G is Hausdorff, we generalize results by Matui and Sato to show that α is cocycle conjugate to α⊗idZ, even if α is not strongly outer. In particular, in this case the equivalences above hold for α in place of α⊗idZ. In the course of the proof, we develop equivariant versions of complemented partitions of unity and uniform property Γ as technical tools of independent interest.

KW - Amenability

KW - Group action

KW - Jiang-Su algebra

KW - Strong outerness

UR - http://www.scopus.com/inward/record.url?scp=85129378310&partnerID=8YFLogxK

U2 - 10.1016/j.matpur.2022.04.003

DO - 10.1016/j.matpur.2022.04.003

M3 - Article

AN - SCOPUS:85129378310

SN - 0021-7824

VL - 162

SP - 76

EP - 123

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

ER -