in order to prove the statement, it suffices to prove existence of a sequence $(w_k)_{k\in{\mathbb{N}}}$ of elements in ${\mathbb{C}}\langle X_1,\dots,X_n\rangle$, which satisfy only conditions (ii) and (iii). Indeed, if we replace in this case $w_k$ by its real part $\Re(w_k)=\frac{1}{2}(w_k+w_k^\ast)$, conditions (ii) and (iii) are still valid, but we have achieved condition (i) in addition.

For proving existence under these weaker conditions, we may apply Kaplansky’s density theorem. This guarantees the existence of a sequence $(w_k)_{k\in{\mathbb{N}}}$ of elements in ${\mathbb{C}}\langle X_1,\dots,X_n\rangle$, such that $\|w_k\| \leq \|w\|$ for all $k\in{\mathbb{N}}$, which particularly implies (ii), and which converges to $w$ in the strong operator topology. It remains to note that, with respect to the weak operator topology, $w_k^\ast w \rightarrow w^\ast w$, $w^\ast w_k \rightarrow w^\ast w$, and $w_k^\ast w_k \rightarrow w^\ast w$ as $k\rightarrow\infty$, such that according to the continuity of $\tau$ $$\begin{aligned}
\|w_k-w\|_2^2 &= \tau((w_k-w)^\ast(w_k-w))\\
              &= \tau(w_k^\ast w_k) - \tau(w_k^\ast w) - \tau(w^\ast w_k) + \tau(w^\ast w)\end{aligned}$$ tends to $0$ as $k\rightarrow 0$, which shows (iii) and thus concludes the proof.

The main result of this subsection is the following proposition. There, we will use the projective norm $\|\cdot\|_\pi$ on ${\mathbb{C}}\langle X_1,\dots,X_n\rangle^{\otimes 2}$, which is given by $$\|Q\|_\pi := \inf\bigg\{\sum^N_{k=1} 