Let $(\mathfrak{M}, \tau)$ be a W${}^{\ast}$-Algebra with (finite, normal, etc., whichever nice conditions one may find need for) tracial state. Elements $(a_{i})_{i\in I}\subset\mathfrak{M}$ shall be called free, exactly in case for all finite collections of polynomials $(p_{k})_{k<n}\subset\mathbb{C}[X]$ and indices $(i_{k})_{k<n}\subseteq I$ with $i_{j}\neq i_{k}$ for $j,k<n$ with $|j-k|=1$, it holds that:
$$.\forall{k<n}:\tau(p_{k}(a_{i_{k}}))=0. ~\Longrightarrow~.\tau(\prod_{k<n}p_{k}(a_{i_{k}}))=0.$$
Question. Let $(x_{i})_{i<N}\subset\mathfrak{M}$ be finitely many elements. Do there exist finitely many free elements $(a_{j})_{j<M}\subset\mathfrak{M}$, so that the $x_{i}$ are somehow (e. g. spectral-theoretically) generated by the $a_{j}$?
Remark 1. of course a desirable consquence, is that the free elements are somehow (e. g. spectral-theoretically) constructable out of the original elements.
Remark 2. It is not necessary that $M=N$ or that even $M\leq N$. An answer for the case $N=2$ would also be useful. One can also always reduce to the case, that all original elements are self-adjoint (resp. positive, resp. unitary).
Remark 3. This question of mine essentially asks, if there be a non-commutative Gram-Schmidt process.