From "Lectures on the combinatorics of free probability" by Nica and Speicher we have a necessary sufficient criteria for an element of $M_n(M)$ being free from $M_n(\mathbb{C})$ for a non-commutative probability space (NCPS) $M$. Do we have a similar result for such an element being $\ast$-free from $M_n(\mathbb{C})$ for a $\ast$-non-commutative probability space $M$? I can't think of even a good necessary condition.

The result that I mentioned above is: Let $x_{i,j} \in$ an NCPS $(M,\phi)$, $i,j=1, \cdots, n$. TFAE:

(1) The matrix $(x_{i,j}$ is free from $M_n(\mathbb{C})$ in $(M_n(M),Tr)$ where $M_n(M) = M_n(\mathbb{C}) \otimes M$ and $Tr=tr \otimes \phi$.

(2) Free cumulants of $\{x_{i,j}\}$ in $(M,\phi)$ are such that only cyclic cumulants $\kappa_m(x_{i(1),i(2)},x_{i(2),i(3)}, \cdots, x_{i(m),i(1)})$ are possibly different from 0, the value depending only on $m$, not the tuple $(i(1), \cdots, i(m))$.

I am also defining a $\ast$-non-commutative probability space $M$ and $\ast$-freeness below:

Def 1- A non-commutative probability space (NCPS) is a couple $(M,\phi)$ where $M$ is a unital algebra and $\phi$ is a unital linear functional on it. $(M, \phi)$ is called a $\ast$-*non-commutative probability space* if $M$ has an involution $\ast$ on it and $\phi$ is positive w.r.t $\ast$. My interest lies in the case where $M$ is a $II_1$ factor and $\phi$ is tracial state. But the result we have is for general NCPS.

Def 2- A family $(A_i)$ of unital $\ast$-subelgebras of $M$ is called free if $\phi(a_1 \cdots a_n)=0$ whenever $n \ge 1, a_j \in A_{i(j)}, \phi(a_j)=0$ and $i(j) \ne i(j+1)$ for all $j=1, \cdots, n$. $(a_i)$, a family of elements of $M$ are called $\ast$-*free* if the family $(alg(1,a_i,a_i^*))$ is free in $M$.