Take the tour ×  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 $$\sup_{n\in \mathbb{N}} \;d_{{\rm BM}}\big(
\mathscr{B}(P_n(E)) ), P_{n^2}(E) \big)=\infty.$$  share|improve
this question I presume you mean $<\infty$ rather than $=\infty$
in your second question. As for the first question, probably
someone computed the GL constant for $B(\ell_p^n)$. That might
give the order of magnitude for the B-M distance you are
seeking. I would look in Tomczak's book but my copy is loaned
out. –  Bill Johnson Oct 6 '12 at 16:46 The following might be
too trivial for what you want, but how about considering th