I would like to prove that the following linear map is a contraction on the $3 \times 3$ complex matrices endowed with the usual operator norm (i.e. the largest singular value norm): $$ \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \mapsto {2\over 3}\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & {a_{11} + a_{22} \over 2} \end{pmatrix}. $$
Numerical simulation shows this is likely to be true, but it is absolutely not clear to me how to prove it. Any idea is welcomed.