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

TL;DR

This paper characterizes when the continuous core of type III$_1$ free product factors is full, linking it to the topology of the $ au$-invariant, and extends the result to related factors.

Contribution

It establishes a precise criterion for fullness of continuous cores in type III$_1$ free product factors based on the $ au$-invariant topology.

Findings

Fullness of the continuous core is equivalent to the $ au$-invariant being the usual topology.

The result applies to free Araki--Woods factors as a special case.

The method extends to full Bernoulli crossed product factors of type III$_1$.

Abstract

We prove that, for any type 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 III$_1$.