A convex structure on sofic embeddings. (English) Zbl 1300.43006
Summary: N. P. Brown [Adv. Math. 227, No. 4, 1665–1699 (2011; Zbl 1229.46041)] introduced a convex-like structure on the set of unitary equivalence classes of unital \(\ast\)-homomorphisms of a separable type \(II_{1} \) factor into \({R}^{\omega} \) (ultrapower of the hyperfinite factor). The goal of this paper is to introduce such a structure on the set of sofic representations of groups. We prove that if the commutant of a representation acts ergodically on the Loeb measure space then that representation is an extreme point.
MSC:
| 43A30 | Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. |
| 46L55 | Noncommutative dynamical systems |
| 37A15 | General groups of measure-preserving transformations and dynamical systems |
Citations:
Zbl 1229.46041References:
| [1] | DOI: 10.1090/S0002-9947-1975-0390154-8 · doi:10.1090/S0002-9947-1975-0390154-8 |
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