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

A characterization of fullness of continuous cores of type III$_1$ free product factors