In this note, we first work out some `bare hands' computations of the most elementary possible free products involving $\mathbb{ C}^2 ~(=\mathbb{ C} \oplus \mathbb{ C} $) and $M_2 ~(= M_2(\mathbb{ C}))$. Using these, we identify all free products $C \ast D$, where $C,D$ are of the form $A_1 \oplus A_2$ or $M_2(B)$; $A_1,A_2,B$ are finite von Neumann algebras, as is $A_1 \oplus A_2$ with the 'uniform trace' given by $tr(a_1, a_2) = 1/2 (tr(a_1) + tr(a_2))\}$ and $M_2(B)$ with the normalized trace given by $tr((b_{ i,j}))=1/2(tr(b_{ 1,1}) + tr(b_{ 2,2}))$. Those results are then used to compute various possible free products involving certain finite dimensional von-Neumann algebras, the free-group von-Neumann algebras and the hyperfinite $II_1$ factor. In the process, we reprove Dykema's result `$R \ast R \cong LF_2$'.
Categories: math.OA