Subfactors of free products of rescalings of a ${\rm II}_1$-factor
Ken Dykema
Let $Q$ be any ${\rm II}_1$-factor. It is shown that any standard lattice can be realized as the standard invariant of a free product of (several) rescalings of $Q$. In particular, if $Q$ has fundamental group equal to the positive reals and if $P$ is the free product of infinitely many copies of $Q$, then $P$ has subfactors giving rise to all possible standard invariants. Similarly, given a ${\rm II}_1$-subfactor $N\subset M$, it is shown there are subfactors $\hat N\subset\hat M$ having the same standard invariant as $N\subset M$ but where $\hat M$, respectively $\hat N$, is the free product of $M$, respectively $N$, with rescalings of $Q$.