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

2023/02/21 by Nawata, Norio
#46L40 #FOS: Mathematics #Operator Algebras (math.OA) #Primary 46L55 #Secondary 46L35

paper · doi:10.48550/arxiv.2302.10550

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\otimesW 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 II₁ factor π_τ_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 π_τ_A(A)^'' and β on π_τ_B(B)^'' are conjugate (resp. cocycle conjugate). We also construct the model actions.

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