Excuse me if the question is not appropriate for Mathoverflow. I havs asked it in math.stackexchange, but did not get any response. And so, I dared to put it here. I am trying to learn free probability from scratch, mostly by myself. I am trying to prove the following result. 



> 
> If $\mu$ and $\nu$ are compactly supported probability measures, then
>  there exists a $C^\*$ probability space $(\mathcal{A}, \phi)$ and self
>  adjoint $a,~b\in \mathcal{A}$ which are free independent and have
>  distributions $\mu$ and $\nu$ respectively.
> 
> 
> 


I was told that this can be easily proved using free products of $\*$-probability spaces. But I can not see how it follows. Can someone give me a line of proof, or point out some place where I can get the result? Advanced thanks for any helps/suggestions. 


If it is not too much already, can someone suggest some reference from where I can study free probability, possibly from a Physics point of view? (I have already read Terrence Tao's blog notes, and I want to move forward).



Existence of free operators, independent and with given distributions
Consider the C$^\*$-algebras $\mathcal A = L^\infty(\mu)$ and $\mathcal B = L^\infty(\nu)$. They are C$^\*$-probability spaces when equipped with the states $\phi(c) = \int c \;\mu$ and $\psi(c) = \int c \; d\nu$.
Then the elements $a = z$ and $b=z$ in $\mathcal A$ and $\mathcal B$ have distributions $\mu$ and $\nu$ respectively.


Voiculescu defined the **reduced free product** of C$^\*$-algebras in his seminal work on free probability
(Operator Algebras and their Connections with Topology and Ergodic Theory
, Lecture Notes in Math. 1132 (1985), Springer Verlag, 556). Though, you can google reduced free product to turn up some more descriptive descriptions. In simple terms this defines a new C$^\*$-probability space $(\mathcal A \*\_r \mathcal B, \phi \*\_r \psi)$ such that $\mathcal A, \mathcal B \subset \mathcal A \*\_r \mathcal B$, $\phi \*\_r\psi|\_\mathcal A = \phi$, $\phi \*\_r \psi|\_\mathcal B = \psi$ and, most importantly, $\mathcal A$ and $\mathcal B$ are freely independent with respect to $\phi \*\_r \psi$. 


Therefore, $a$ and $b \in \mathcal A \*\_r\mathcal B$ are freely independent with distributions $\mu$ and $\nu$. 


Here are some references, though I am not sure if any deal with a physics perspective:


(1) G. Anderson, A. Guionnet, O. Zeitouni, *An introduction to random matrices.*


(2) A. Nica, R. Speicher: *Lectures on the Combinatorics of Free Probability*. 


(3) F. Hiai and D. Petz, *The Semicircle Law, Free Random Variables, and Entropy.*


(4) P. Mitchener, *Non-Commutative Probability Theory.*


(5) D. Voiculescu, K. Dykema, A. Nica, *Free random variables.* 


