Some explicit computations and models of free products

Madhushree Basu

paper

Some explicit computations and models of free products

arXiv:1111.6183

Abstract

In this note, we first work out some `bare hands' computations of the most elementary possible free products involving $C^{2} (= C \oplus C$ ) and $M_{2} (= M_{2} (C))$ . Using these, we identify all free products $C * D$ , 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 $t r (a_{1}, a_{2}) = 1/2 (t r (a_{1}) + t r (a_{2}))}$ and $M_{2} (B)$ with the normalized trace given by $t r ((b_{i, j})) = 1/2 (t r (b_{1, 1}) + t r (b_{2, 2}))$ . 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 $I I_{1}$ factor. In the process, we reprove Dykema's result ` $R * R ≅ L F_{2}$ '.