freeness []

This page is read only. You can view the source, but not change it. Ask your administrator if you think this is wrong.
========= Freeness =========
----


<typo font-size:medium;>**Definition** 
</typo>[(:cite:freeness:voiculescu1987multiplication)]


Let $\mathcal{A}$ be a unital complex algebra equipped with a linear functional $\psi:\mathcal{A}\to \mathbb{C}$ where $\psi$ is unital. Let $\mathcal{A}_1 ,\mathcal{A}_2 \subset \mathcal{A}$ be two unital subalgebras. The two subalgebras $\mathcal{A}_1$ and $\mathcal{A}_2$ are //free// with respect to $\psi$ if, for any sequence $(a_1,\ldots,a_n)$ of $\mathcal{A}_1\cup \mathcal{A}_2$ which is alternating and centered with respect to $\psi$, we have $$\label{eqn:freemoments}
    \psi(a_1\cdots a_n) = 0.$$


===== - Models =====




Let $R > 0$ a real number. For each $N\geq 1$, let $U_N$ be a Haar distributed unitary random matrix. Then, for any $P\in \mathbb{C}\langle X_k,Y_k:k\in K\rangle$, we have $$\mathbb{E}\left[\frac{1}{N}{\rm Tr}(P({\bf A}_N,U_N{\bf B}_NU_N^{\star}))\right] = \psi_{{\bf A}_N} * \psi_{{\bf B}_N}(P)+O(N^{-2})$$ uniformly for the the choice of any sequences ${\bf A}_N = \{A_N^k: k \in K\}$ and ${\bf B}_N = \{B^k_N:~k \in K\}$ of $N\times N$ matrices ($N\geq 1$) bounded in operator norm by $R$