We call a finite vN algebra $M$ with trace $\tau$ has property Gamma iff there exist unitary $u_{n}\rightarrow 0$ weakly and $\|xu_{n}-u_{n}x\|\rightarrow 0$. My question can we have C*-simple countable discrete $G$ such that $L(G)$ has property Gamma. If it has what are $u_{n}$ in that case?
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$\begingroup$ Yes, e.g. certain Baumslag-Solitar groups. See section 6.3 in the paper: arxiv.org/abs/1410.2518 for discussion of the C$^*$-simplicity. For Gamma property, see MO question 96586. $\endgroup$– JiangCommented Nov 12, 2019 at 14:57
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$\begingroup$ @Jiang Why not post that as an answer, so that the answer can be accepted, and this question can be taken off the "unanswered" list? $\endgroup$– Yemon ChoiCommented Nov 28, 2019 at 15:16
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