Let $\ell\_p^n$ be the $n$-dimensional real or complex $\ell\_p$-space and let $\mathscr{B}(\ell\_p^n)$ be the space of matrices on $\ell\_p^n$ endowed with the operator norm. I am looking for any references which give (reasonable) estimates for the Banach-Mazur distance between $\mathscr{B}(\ell\_p^n)$ and $\ell\_p^{n^2}$.


I am also wondering whether there is a Banach space $E$ with a Schauder basis $(e\_n)$ and corresponding basis projections $(P\_n)$ which satisfies
$$\sup\_{n\in \mathbb{N}} \;d\_{{\rm BM}}\big( \mathscr{B}(P\_n(E)) ), P\_{n^2}(E) \big)=\infty.$$



BM-distances between $B(\ell_p^n)$ and $\ell^{n^2}_p$
For the first question, $n^{1/2}$ is the right order for $p=1,2,\infty$. The case $p=\infty$ is arguably the easiest, because $B(\ell\_\infty^n)=\ell\_\infty^n(\ell\_1^n)$ isometrically and the Banach-Mazur distance between $\ell\_\infty^n$ and $\ell\_1^n$ is of order $n^{1/2}$. That gives the upper bound, and the lower bound is also true because $\gamma\_\infty(\ell\_1^n)$ (the factorization constant of the identity on $\ell\_1^n$ through an $\ell\_\infty$-space) is of order $n^{1/2}$.  

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Also $B(\ell\_1^n)=\ell\_\infty^n(\ell\_1^n)$ isometrically, and you get the lower bound from the fact that $\gamma\_1(\ell\_\infty^n)$ is of order $n^{1/2}$. For the upper bound assume that $n$ is a power of two (I think standard arguments from this case give the general case but did not try to think it through). In $L\_1^N(\ell\_1^n)$ consider the basis $w\_k\otimes e\_j$, $1\le j,k \le n$, where $(w\_k)$ is the Walsh basis for $L\_1^n$. Take the basis to basis mapping from $L\_1^N(\ell\_1^n)$ onto $\ell\_\infty^n(\ell\_1^n)$ that maps $w\_k\otimes e\_j$, $1\le j \le n$, onto the $k$-th copy of $\ell\_1^n$. This mapping has norm at most $n^{1/2}$ because the Walsh basis has an upper $\ell\_2$ estimate, and the inverse has norm one.
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For $B(\ell\_2^n)$, Yemon's comment gives a lower estimate of $n^{1/2}$ (in fact, for all $1\le p \le 2$ because $\gamma\_p(\ell\_\infty^n)$ is of order $n^{1/2}$ in this range). The upper estimate is just the fact that the norm of an operator in $\ell\_2^n$ is at most $n^{1/2}$ times the Hilbert-Schmidt norm of the operator.
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For all values of $p$, if you write a vector in $\mathbb{R}^{n^2}$ as an $n$ by $n$ matrix $A$, then the $\ell\_p^{n^2}$ norm of $A$ is the $p'$-summing norm of $A$ considered as an operator from $\ell\_p^n$ into $\ell\_{p'}^{n}$. That is what is used above in the $p=2$ case, but I don't see that this helps for other values of $p$.
For $2<p<\infty$ you get a lower bound from the fact that $B(\ell\_p^n)$ contains $\ell\_\infty^n$ and $\ell\_{p'}^n$ isometrically, but for $p=4$ that gives a lower bound of only $n^{1/4}$.


