Strongly outer actions of certain torsion-free amenable groups on the Razak-Jacelon algebra
Abstract
Let $\mathfrak{C}$ be the smallest class of countable discrete groups with the following properties: (i) $\mathfrak{C}$ contains the trivial group, (ii) $\mathfrak{C}$ is closed under isomorphisms, countable increasing unions and extensions by $\mathbb{Z}$. Note that $\mathfrak{C}$ contains all countable discrete torsion-free abelian groups and poly-$\mathbb{Z}$ groups. Also, $\mathfrak{C}$ is a subclass of the class of countable discrete torsion-free elementary amenable groups. In this paper, we show that if $\Gamma\in \mathfrak{C}$, then all strongly outer actions of $\Gamma$ on the Razak-Jacelon algebra $\mathcal{W}$ are cocycle conjugate to each other. This can be regarded as an analogous result of Szabó's result for strongly self-absorbing C$^*$-algebras.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2023
- DOI:
- 10.48550/arXiv.2309.00934
- arXiv:
- arXiv:2309.00934
- Bibcode:
- 2023arXiv230900934N
- Keywords:
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- Mathematics - Operator Algebras;
- Primary 46L55;
- Secondary 46L35;
- 46L40
- E-Print:
- 8 pages, I have changed Definition 2.2