Structural results for free Araki–Woods factors and their continuous cores

Abstract

AbstractWe show that for any type III1free Araki–Woods factor$\mathcal{M}$=(HR, Ut)″ associated with an orthogonal representation(Ut)ofRon a separable real Hilbert spaceHR, the continuous coreM=$\mathcal{M}$⋊σRis a semisolid II∞factor, i.e. for any non-zero finite projectionq∈M, the II1factorqM qis semisolid. If the representation(Ut)is moreover assumed to be mixing, then we prove that the coreMis solid. As an application, we construct an example of a non-amenable solid II1factorNwith full fundamental group, i.e.$\mathcal{F}$(N) =R*+, which is not isomorphic to any interpolated free group factorL(Ft), for 1 <t≤ = +∞.