Some Explicit Computations and Models of Free Products

In this note, we first work out some ‘bare hands’ computations of the most elementary possible free products involving C² (= C ⊕ C), with the ‘uniform trace’ given by tr (z₁, z₂) = 1/2 (tr (z₁) + tr (z₂)), and M₂ (= M₂(C)), with the normalized trace of the matrix. Using these, we identify all free products of the form C ∗ D where {C, D} ⊂ {A₁ ⊕ A₂, M₂(B)}, where A_i , B are finite von Neumann algebras, as is A₁ ⊕ A₂, with the trace tr (a₁, a₂) = 1/2 (tr (a₁) + tr (a₂)) and M₂(B) (∼= M₂⊗ B) with the trace tr M₂ ⊗tr_B. 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 I I1 factor. In the process, we reprove Dykema’s result ‘R ∗ R ∼= LF₂’.