An example for almost commuting matrices I'm looking for examples of the following kind:
2 sequences of unitary matrices ($A_n,B_n$) (where these are $n\times n$ matrices) that don't commute and satisfy
$$
||A_nB_n-B_nA_n||_1\longrightarrow_{n\rightarrow\infty}0
$$
Where the $||\cdot||_1$ norm is the sum of the absolute values of entries.
I know that for other norms (actually for any $p>1$, there exists an example through a permutation matrix and a sign matrix - for example the right shift and the ordered roots of unity. This unfortunately doesn't work for the absolute value norm so I'm stuck)
I'd appreciate any help, thank you.
 A: You could take rotation matrices in $\mathbb{R}^3$ which usually do not commute. If you define the matrices in such way that the angle of rotation becomes smaller when $n$ gets larger, the matrices will approach the identity matrix for increasing $n$ and the norm of the commutator will approach $0$, independently of the choice of the particular norm. Example:
$$
A_n = \begin{pmatrix}
\cos\frac{\pi}{n} & -\sin\frac{\pi}{n} & 0 \\
\sin\frac{\pi}{n} & \cos\frac{\pi}{n} & 0 \\
0  & 0 & 1
\end{pmatrix}
$$
and
$$
B_n = \begin{pmatrix}
 1 & 0 & 0 \\
0 & \cos\frac{\pi}{n} & -\sin\frac{\pi}{n} \\
0 & \sin\frac{\pi}{n} & \cos\frac{\pi}{n} \\
\end{pmatrix}
$$
A: Take 
$$
A_n=\begin{bmatrix}\cos\tfrac1n&\sin\tfrac1n\\
\sin\tfrac1n&-\cos\tfrac1n\\
&&1\\
&&&\ddots&\\
&&&&1
\end{bmatrix},
\ \ \ \ \ 
B_n=\begin{bmatrix}-1&\\
&1\\
&&1\\
&&&\ddots&\\
&&&&1
\end{bmatrix}.
$$
Then
$$
\|A_nB_n-B_nA_n\|_1=\left\|\begin{bmatrix}0&2\sin\tfrac1n\\-2\sin\tfrac1n&0\end{bmatrix}\right\|_1=4\,\left|\sin\tfrac1n\right|\leq\tfrac4n
$$
