Given two free semicirculars X\_1 and X\_2 and a projection h in the von-Neumann algebra generated by X\_1, how does one show that the von-Neumann algebra generated by {X\_1, hX\_2(1-h)} is a factor? It is easy to show that the two elements in the generating set are free. But I am unable to see what kind of an object hX\_2(1-h) is. It appears in the definition of interpolated free group factor in Radulescu's paper (pre-print 1991) on random matrices, amalgamated free products and subfactors of free group factors of non-integer index. 



v-Na generated by 
Let me first point out that $X\_1$ and $Y=h X\_2 (1-h)$ are *not* freely independent. This is most easily seen if $h$ has trace 1/2, in which case $Y$ has range and support projections $h$ and $(1-h)$, respectively. But since the support and range projections of $Y$ belong to $W^\*(Y)$, it would follow from the assumption that $Y$ and $X$ are free that actually $h$ and $X$ are free. But this is not possible, since they commute.


Now to your question of factoriality. Here is a sketch of the proof. Let us assume for definiteness that $\tau(h) \geq \tau (1-h)$ (otherwise, switch $Y$ and $Y^\ast$). You can then verify that $Y^\ast Y$ and $YY^\ast$ have free Poisson distributions (with different parameters) and that the spectrum of $YY^\ast$ has no atoms. It follows that if you consider the polar decomposition of $Y = V |Y|$, then $V$ is a partial isometry with domain projection $1-h$ and range projection $\leq h$. Using this, you can see that $W^\*(X\_1,Y)$ is a factor iff $N=h W^\*(X\_1,Y)h $ is a factor. But $N$ is generated by $hX\_1h$ and $Y^\ast Y$; you can prove that these elements are freely independent (in $N$). This either uses a random matrix model (see e.g. Voiculescu's book on free random variables for the proof of the compression formula for free group factors), or can be done directly using operator-valued semicircular systems. Thus $N = W^\*(hX\_1 H) \* W^\*(Y^\ast Y)$ which is a free product of two abelian von Neumann algebras, one of which is diffuse and the other not complex numbers. You can then get factoriality (see references in Ueda's paper <http://arxiv.org/abs/1011.5017>)


