Let $(A,\varphi)$ be a $^*$-probability space, i.e.,
- $A$ is a unital $^*$-algebra, and
- $\varphi$ is a $^*$-preserving unital linear functional.
Let $(c_{i,j})_{i,j\in I}$ be a positive definite matrix.
A centered semicircular family in $(A,\varphi)$ of covariance $(c_{i,j})_{i,j\in I}$ is a family $(s_i)_{i\in I}$ of selfadjoint elements in $A$ such that
- each $s_i$ is centered, i.e., $\varphi(s_i) = 0$ for each $i\in I$, and
- the distribution of $(s_i)_{i\in I}$ satisfies \begin{equation*} \varphi(s_{i(1)}\cdots s_{i(m)}) = \sum_{\pi\in\text{NC}_2(m)}\kappa_{\pi}(s_{i(1)},\ldots,s_{i(m)}) \end{equation*} for all positive integers $m$ and $i(1),\ldots,i(m)\in I$, where $\text{NC}_2(m)$ is the set of all noncrossing pair (i.e., blocks of size 2) partitions of $\{1,\ldots,m\}$, and $\kappa_{\pi}(s_{i(1)},\ldots,s_{i(m)})$ is a free cumulant.
See e.g. Lectures on the Combinatorics of Free Probability by Nica and Speicher, page 128. The definition of free cumulants can also be found there (page 175): \begin{equation*} \kappa_{\pi}(a_1,\ldots,a_m) = \sum_{\substack{ \sigma\in\text{NC}(m)\\ \sigma\leq\pi}} \mu(\sigma,\pi) \prod_{V\in\sigma} \varphi\bigg( \prod_{i\in V}a_i \bigg) \end{equation*} for any positive integer $m$ and any $a_1,\ldots,a_m\in A$ and any noncrossing partition $\pi$ of $\{1,\ldots,m\}$, where $\mu$ is the Mobius function on the lattice $\text{NC}(m)$ of noncrossing partitions of $\{1,\ldots,m\}$.
The first condition of each $s_i$ being centered is no issue: each $s_i$ can be made centered by substracting its mean $\varphi(s_i)$. But what about the second one? Does there always exist a centered family $(s_i)_{i\in I}$ satisfying \begin{equation*} \varphi(s_{i(1)}\cdots s_{i(m)}) = \sum_{\pi\in\text{NC}_2(m)}\kappa_{\pi}(s_{i(1)},\ldots,s_{i(m)}) \end{equation*} for all positive integers $m$ and $i(1),\ldots,i(m)\in I$?
This second condition is very similar to the free moment-cumulant formula, with the only difference that in the latter the sum runs over all noncrossing partitions, not just the ones whose blocks are sets of cardinality 2: \begin{equation*} \varphi(a_1\cdots a_m) = \sum_{\pi\in\text{NC}(m)}\kappa_{\pi}(a_1,\ldots,a_m) \end{equation*} for all positive integers $m$ and elements $a_1,\ldots,a_m\in A$, where $\text{NC}(m)$ is the set of all noncrossing partitions of $\{1,\ldots,m\}$.
The second condition will in particular be satisfied if all cumulants $\kappa_{\pi}(s_{i(1)},\ldots,s_{i(m)})$ vanish in case the noncrossing partition $\pi$ is not a pair partition. This is of course a stronger condition. If $\pi$ is a noncrossing partition containing a block with just one element $j$, then it vanishes as a result of $s_{i(j)}$ being centered: free cumulants are multiplicative, in the sense that $\kappa_{\pi}(a_1,\ldots,a_m)$ for any $a_1,\ldots,a_m\in A$ factorizes over the blocks $V$ in $\pi$, i.e., \begin{equation*} \kappa_{\pi}(a_1,\ldots,a_m) = \prod_{V\in\pi} \kappa_{\{V\}}(a_k:k\in V), \end{equation*} where each $\kappa_{\{V\}}(a_k:k\in V)$ is the free cumulant restricted to the partition consisting of just the block $V$, and the free cumulant $\kappa_{\{j\}}(a)$ for any $a\in A$ and $j\in\{1,\ldots,m\}$ is just $\varphi(a)$, so $\kappa_{\{j\}}(s_{i(j)}) = \varphi(s_{i(j)}) = 0$. But this still leaves the noncrossing partitions $\pi$ with no blocks of size 1 and at least one block of size greater than 2. What about their free cumulants? Are there any families $(s_i)_{i\in I}$ for which these free cumulants vanish, making $(s_i)_{i\in I}$ a semicircular family?
In the case where the index set $I$ consists of just one element, the definition reduces to that of a centered semicircular element, which is no issue. So what is of interest is the case where $I$ has cardinality greater than one.
Yes, such semicircular families exist. In fact, they can be represented by bounded operators. This construction came at the very beginning of free probability theory.
Let $H$ be a real Hilbert space with vectors $\xi_i\in H$ such that $\langle\xi_i,\xi_j\rangle=c_{i,j}$. Define $\mathcal F(H)=\mathbb C\Omega\oplus \bigoplus_{n=1}^\infty H_\mathbb C^{\otimes n}$, where $\Omega$ is some fixed unit vector and $H_{\mathbb C}$ is the complexification of $H$. For $\xi\in H$ define $$ a(\xi)\Omega=\xi,\quad a(\xi)(\eta_1\otimes\dots\otimes\eta_n)=\xi\otimes \eta_1\otimes\dots\otimes\eta_n\\ a^\ast(\xi)\Omega=0,\quad a^\ast(\xi)(\eta_1\otimes\dots\otimes\eta_n)=\langle\xi,\eta_1\rangle\eta_2\otimes\dots\otimes\eta_n. $$ These operators extend to bounded linear operators on $\mathcal F(H)$, which are adjoint to each other. Let $s(\xi)=a(\xi)+a^\ast(\xi)$ and $\phi=\langle\Omega,\cdot\,\Omega\rangle$.
Note that $\phi(s(\xi_i))=\langle\Omega,\xi_i\rangle=0$ and $$ \phi(s(\xi_i)s(\xi_j))=\langle s(\xi_i)\Omega,s(\xi_j)\Omega\rangle=\langle\xi_i,\xi_j\rangle=c_{i,j}. $$
For the joint moments, we have $$ \phi(s(\xi_{j(1)})\dots s(\xi_{j(n)}))=\langle\Omega,(a(\xi_{j(1)})+a^\ast(\xi_{j(1)}))\dots (a(\xi_{j(n)})+a^\ast(\xi_{j(n)}))\Omega\rangle. $$ The creation operator $a(\xi)$ maps $H^{\otimes n}$ to $H^{\otimes(n+1)}$, i.e., one step up, and the annihilation operator $a^\ast(\xi)$ maps $H^{\otimes n}$ to $H^{\otimes(n-1)}$, i.e., one step down. Hence, if we expand the sum on the right side, we only get a non-zero contribution if the number of annihilation and creation operators in the product is the same. In particular, the joint moment is zero if $n$ is odd. Moreover, the first factor has to be an annihilation operator and the last factor has to be a creation operator to get non-zero contributions.
Now, products of this form can be reduced using the identity $a^\ast(\xi)a(\eta)=\langle \xi,\eta\rangle \mathrm{id}$, which allows to show inductively $$ \phi(s(\xi_{j(1)})\dots s(\xi_{j(n)}))=\sum_{\sigma\in \mathrm{NC}_2(n)}\prod_{\substack{\{k,l\}\in\sigma\\k<l}}\langle\xi_{j(k)},\xi_{j(l)}\rangle. $$ This "taking steps up and down" is nicely illustrated by (rotated) Dyck paths, which are closely related to noncrossing pair partitions.
As a reference, this is all pretty nicely explained in Section 6.1 here: https://www.math.tamu.edu/~manshel/m689/Free-probability-notes.pdf.