A characterization of the fullness of continuous cores of type III 1 free product factors

Abstract

We prove that, for any type ${\mathrm{III}}_1$ free product factor, its continuous core is full if and only if its $\tau$-invariant is the usual topology on the real line. This trivially implies, as a particular case, the same result for free Araki–Woods factors. Moreover, our method shows the same result for full (generalized) Bernoulli crossed product factors of type ${\mathrm{III}}_1$.