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Spectral gap actions and invariant states
- Publication Year :
- 2013
-
Abstract
- We define spectral gap actions of discrete groups on von Neumann algebras and study their relations with invariant states. We will show that a finitely generated ICC group $\Gamma$ is inner amenable if and only if there exist more than one inner invariant states on the group von Neumann algebra $L(\Gamma)$. Moreover, a countable discrete group $\Gamma$ has property $(T)$ if and only if for any action $\alpha$ of $\Gamma$ on a von Neumann algebra $N$, every $\alpha$-invariant state on $N$ is a weak-$^*$-limit of a net of normal $\alpha$-invariant states.<br />Comment: 10 pages; to appear in Int. Math. Res. Notices
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1304.7051
- Document Type :
- Working Paper