The standard polynomial in $r$ non-commuting indeterminates $x_1,\ldots,x_r$ is defined by $${\mathcal S}_r(x_1,\ldots,x_r):=\sum_{\sigma\in S_r}\epsilon(\sigma)x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\sigma(r)}\,$$ where $S_r$ is the symmetric group in $r$ letters and $\epsilon$ is the signature. Each monomial is a word in the letters $x_j$, affected of a sign $\pm1$. Finally, let us normalize $$T_r:={1\over r!}{\mathcal S}_r.$$
When $r=2$, ${\mathcal S}_2(x_1,x_2)=x_1x_2-x_2x_1$ is the commutator. When using the Frobenius norm $\|M\|=\sqrt{Tr(M^*M)}$ , we know from [1] that $\|[A,B]\|\le\sqrt2\|A\|\cdot\|B\|$, and $\sqrt2$ is the least constant in this inequality, whatever the dimension $n\ge2$.
If $r\ge2$, what is the least constant $c(r,n)$ in the inequality $$\|T_r(A_1,\ldots,A_r)\|\le c(r;n)\prod_{j=1}^r\|A_j\|,\quad\forall A_1,\ldots,A_r\in M_n(\mathbb{C})?$$
Because $n\mapsto c(r,n)$ is non-decreasing, we set $\tau(r)=\lim_{n\rightarrow+\infty}=\sup_n c(r,n)$.
By Amitsur--Levitski Theorem (see [2] for a short proof, or Section 4.4 of the second edition of my book on Matrices), we have $c(r,n)=0$ when $r\ge2n$. From above, we have $c(2,n)=\sqrt2/2$. I can also prove the inequalities $$c(r+s,n)\le c(r,n)\cdot c(s,n),\qquad \tau(r+s)\le\tau(r)\tau(s).$$ In particular, the limit $$\rho:=\lim_{r\rightarrow+\infty}\tau(r)^{1/r}$$ exists and is also the infimum of the sequence. What is its value ?
Finally, is there a quantitative counterpart of Amitsur--Levitski, in the sense that $r\mapsto c(r,n)$ would drop to a small value at some threshold value, less than $2n$ ?
[1] Seak-Weng Vong, Xiao-Qing Jin, Proof of Böttcher and Wenzel’s conjecture. Oper. Matrices, 2 (2008), pp 435--442.
[2] S. Rosset. A new proof of the Amitsur-Levitski identity. Israel J. Math., 23 (1976), pp 187--188.
