# Morphism of von Neumann Algebras

Hello,

Is there a counterexample to the following statement: let $A,B$ two von Neumann algebras, every morphism $A \rightarrow B$ of $C^*$-algebras is a $W^*$-homomorphism ?

( a $W^*$-homomorphism is a continuous morphism for the weak topologies $\sigma(A,A_* )$ and $\sigma(B,B_* )$, where $A_*$ and $B_*$ are the preduals)

Take $B$ to be the bidual of $A$ and take your morphism to be the canonical inclusion of $A$ into $A^{**}$. – Yemon Choi Sep 20 '11 at 23:08
It might be worth adding that any C*-isomorphism between $A$ and $B$, if there is one, is also a W*-isomorphism. – Nik Weaver Nov 20 '12 at 23:39