A Note about proving non-$Γ$ under a finite non-microstates free Fisher information Assumption

Details A Note about proving non-$Γ$ under a finite non-microstates free Fisher information Assumption A Note about proving non-$Γ$ under a finite non-microstates free Fisher information Assumption

2008-04-18

arxiv.org/abs/0804.2906v2

Journal of Functional Analysis Volume 258, Issue 11, June 2010, Pages 3662-3674

Mathematics

Operator Algebras

12 pages; New results with similar techniques; cf. abstracts for details

Scientific paper

10.1016/j.jfa.2010.02.010

We prove that if $X_{1},...,X_{n} (n >1)$ are selfadjoints in a $W^{*}$-probability space with finite non-microstates free Fisher information, then the von Neumann algebra $W^{*}(X_{1},...,X_{n})$ they generate doesn't have property $\Gamma$ (especially is not amenable). This is an analog of a well-known result of Voiculescu for microstates free entropy. We also prove factoriality under finite non-microstates entropy.

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