Skip to main content Accessibility help
×
Home

Fullness of crossed products of factors by discrete groups

  • Amine Marrakchi (a1)

Abstract

Let M be an arbitrary factor and $\sigma : \Gamma \curvearrowright M$ an action of a discrete group. In this paper, we study the fullness of the crossed product $M \rtimes _\sigma \Gamma $ . When Γ is amenable, we obtain a complete characterization: the crossed product factor $M \rtimes _\sigma \Gamma $ is full if and only if M is full and the quotient map $\overline {\sigma } : \Gamma \rightarrow {\rm out}(M)$ has finite kernel and discrete image. This answers the question of Jones from [11]. When M is full and Γ is arbitrary, we give a sufficient condition for $M \rtimes _\sigma \Gamma $ to be full which generalizes both Jones' criterion and Choda's criterion. In particular, we show that if M is any full factor (possibly of type III) and Γ is a non-inner amenable group, then the crossed product $M \rtimes _\sigma \Gamma $ is full.

Copyright

References

Hide All
1Ando, H. and Haagerup, U.. Ultraproducts of von Neumann algebras.. J. Funct. Anal. 266 (2014), 68426913.
2Choda, M.. Inner amenability and fullness. Proc. Amer. Math. Soc. 86 (1982), 663666.
3Connes, A.. Almost periodic states and factors of type III1. J. Funct. Anal. 16 (1974), 415445.
4Connes, A.. Classification of injective factors. Cases II1, II, IIIλ, λ ≠ 1. Ann. of Math. 74 (1976), 73115.
5Duchesne, B., Tucker-Drob, R. and Wesolek, P.. CAT(0) cube complexes and inner amenability. arXiv: 1903.01596.
6Haagerup, U.. The standard form of von Neumann algebras. Math. Scand. 37 (1975), 271283.
7Houdayer, C.. Structure of II1 factors arising from free Bogoljubov actions of arbitrary groups. Adv. Math. 260 (2014), 414457.
8Houdayer, C. and Isono, Y.. Bi-exact groups, strongly ergodic actions and group measure space type III factors with no central sequence. Comm. Math. Phys. 348 (2016), 9911015.
9Houdayer, C. and Isono, Y.. Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras. To appear in Proc. Roy. Soc. Edinburgh Sect. A.
10Houdayer, C. and Trom, B.. Structure of extensions of free Araki-Woods factors. Preprint. arXiv:1812.08478.
11Jones, V. F. R.. Central sequences in crossed products of full factors. Duke Math. J. 49 (1982), 2933.
12Marrakchi, A.. Spectral gap characterization of full type III factors. To appear in J. Reine Angew. Math. arXiv: 1605.09613.
13Marrakchi, A.. Strongly ergodic actions have local spectral gap. Proc. Amer. Math. Soc. 146 (2017), 38873893.
14Murray, F. and von Neumann, J.. Rings of operators. IV. Ann. of Math. 44 (1943), 716808.
15Ocneanu, A.. Actions of discrete amenable groups on von Neumann algebras. Lecture Notes in Mathematics,vol. 1138 (Berlin: Springer-Verlag, 1985). iv+115 pp.
16Ozawa, N.. A remark on fullness of some group measure space von Neumann algebras. Compos. Math. 152 (2016), 24932502.
17Popa, S., Shlyakhtenko, D. and Vaes, S.. Classification of regular subalgebras of the hyperfinite II1 factor. Preprint.
18Vaes, S. and Verraedt, P.. Classification of type III Bernoulli crossed products. Adv. Math. 281 (2015), 296332.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed