Some explicit computations and models of free products

TL;DR

This paper performs explicit computations of free products involving finite-dimensional von Neumann algebras, identifying their structures and rederiving known results like the free group factor from free products of hyperfinite factors.

Contribution

It provides concrete calculations of free products of specific finite von Neumann algebras and extends understanding of their structure, including reproofs of established results.

Findings

Identified free products of certain finite-dimensional von Neumann algebras.

Computed free products involving free-group von Neumann algebras and hyperfinite factors.

Reproved Dykema's result that R * R is isomorphic to LF_2.

Abstract

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 `$R \ast R \cong LF_2$'.