Morphism of algebras I would like to find some morphisms of $\mathbb C$-algebras $ \Phi : \mathcal{M}_3 ( \mathbb{C} ) \to \mathcal{M}_3 ( \mathbb{C} ) $  such that 
 $$ \ \ \Phi( M ) \ = \ \Phi( J^{-1} M J ) \ = \ \Phi ( JMJ^{-1} ),\ \forall M \in \mathcal{M}_3 ( \mathbb{C} ) ,$$ knowing that $ J = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} $. 
Thanks a lot for your help.
 A: There are no such morphisms. The algebra $\mathcal{M}_3(\Bbb{C})$ has no non-trivial ideals. The kernel of a morphism of algebras is always a proper ideal. Therefore the kernel of any ring homomorphism $\Psi:\mathcal{M}_3(\Bbb{C})\to R$, where $R$ can be any ring, must be trivial.
But you require that $\Phi(JMJ^{-1})=\Phi(M)$. This means that $JMJ^{-1}-M$ is in the kernel of $\Phi$. But $J$ is not in the center of the matrix algebra, so $JMJ^{-1}-M\neq0$ for some matrices $M$. This proves the non-existence.
A: Choose your favorite invertible matrix $A$ that commutes with both $J$ and $J^{-1} = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}$. Then set $\Phi(M) = A^{-1}MA$. Since automorphisms of matrix algebras are inner (http://en.wikipedia.org/wiki/Skolem%E2%80%93Noether_theorem), that's your only option. 
Now, notice that, for a generic matrix $A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$, we have that $$J^{-1}AJ = \begin{pmatrix} a_{31} & a_{32} & a_{33} \\ a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix}J = \begin{pmatrix} a_{33} & a_{31} & a_{32} \\ a_{13} & a_{11} & a_{12} \\ a_{23} & a_{21} & a_{22} \end{pmatrix}.$$ (That's exactly what we get if we apply the permutation $(132)$, i.e. $J$, to the indices of the entries.) Now, for that to equal $A$, we need $a_{11} = a_{33} = a_{22}$, $a_{12} = a_{31} = a_{23}$, and $a_{13} = a_{32} = a_{21}$. A similar argument shows that's enough for it to commute with $J^{-1}$ as well. So choose a circulant matrix $A= \begin{pmatrix} a & b & c \\ c & a & b \\ b & c & a \end{pmatrix}$ and conjugate by it. Those are all the automorphisms that could work.
