Let $I$ be a set, and for each $i \in I$, let $(A_i,\tau_i)$ be an algebraic noncommutative probability space; that is, $A_i$ is a unital complex algebra and $\tau_i: A_i \to \mathbb{C}$ is a unital linear functional. For each $i \in I$, set $A_i^\circ := \ker \tau_i$. Then there exist an algebraic noncommutative probability space $(A,\tau)$ and unital algebra homomorphisms $\iota_i: A_i \to A$ for $i \in I$ such that $A$ may be chosen to be the unital algebraic free product $*_{i \in I} A_i$, amalgamated over the common unit, with the standard reduced-word vector-space decomposition

\begin{align*} A = \mathbb{C}1_A \oplus \bigoplus_{n=1}^{\infty} \bigoplus_{i_1,\dots,i_n \in I,\ i_k \neq i_{k+1}} A_{i_1}^\circ \otimes \cdots \otimes A_{i_n}^\circ, \end{align*}

where the inner direct sum is over finite words with adjacent indices distinct. The pair $(A,\tau)$ and the maps $(\iota_i)_{i \in I}$ satisfy:

1. $\tau(\iota_i(a)) = \tau_i(a)$ for every $i \in I$ and every $a \in A_i$. 2. The subalgebras $\iota_i(A_i) \subset A$ are freely independent in $(A,\tau)$; equivalently, whenever $n \in \mathbb{N}$, $i_1,\dots,i_n \in I$ satisfy $i_k \neq i_{k+1}$ for $1 \leq k < n$, and $a_k \in A_{i_k}$ satisfy $\tau_{i_k}(a_k)=0$, one has $\tau(\iota_{i_1}(a_1)\cdots \iota_{i_n}(a_n)) = 0$.