Approximate representability of finite abelian group actions on the Razak-Jacelon algebra

Abstract

Let A be a simple separable nuclear monotracial C * -algebra, and let α be an outer action of a finite abelian group Γ on A. In this paper, we show that α ⊗ id W on A ⊗ W is approximately representable if and only if the characteristic invariant of α is trivial, where W is the Razak-Jacelon algebra and α is the induced action on the injective II 1 factor πτ A (A)′′ . As an application of this result, we classify such actions up to conjugacy and cocycle conjugacy. In particular, we show the following: Let A and B be simple separable nuclear monotracial C * -algebras, and let α and β be outer actions of a finite abelian group Γ on A and B, respectively. Assume that the characteristic invariants of α and β are trivial. Then α ⊗ id W and β ⊗ id W are conjugate (resp. cocycle conjugate) if and only if α on πτ A (A) ′′ and β on πτ B (B) ′′ are conjugate (resp. cocycle conjugate). We also construct the model actions.