Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\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 `$R \ast R \cong LF_2$'.

31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin
18
The categories of this paper are: econ
16
The categories of this paper are: stat
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
37
The categories of this paper are: supr-con
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
24
The categories of this paper are: dg-ga
19
The categories of this paper are: eess