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

Eusebio Gardella, Ilan Hirshberg, Andrea Vaccaro

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)76-123
Number of pages48
JournalJournal des Mathematiques Pures et Appliquees
StatePublished - 1 Jun 2022


  • Amenability
  • Group action
  • Jiang-Su algebra
  • Strong outerness

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics


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