Non spatial isomorphisms Let $H$ be a Hilbert space. Any unitary operator $u\in B(H)$ induces an spatial isomorphism $\phi_u(x)=uxu^*$ on $B(H)$. 
Question: Let $\phi:B(H)\to B(H)$ be a surjective  *-ismorphism. Is $\phi$ implemented by a unitary operator $u\in B(H)$ i,e. $\phi=\phi_u$?
Remark. As for this problem, let us review two facts: 
i) It is well-known that any surjective *-isomorphism $\phi$ on $B(H)$ (and in general on any von Neumann algebra) is $w^*$-continuous. 
ii) Let $\psi$ be the restriction of $\phi$ on compact operators $K(H)$. Assume  $\psi$ forms an isomorphism from $K(H)$ onto $K(H)$ then $\psi$ is implemented by a unitary operator and then i) implies that $\phi$ is also implemented by a 
unitary operator.
This remark says that, my problem may be re-written in this form too: Does there exist any surjective $*$-isomorphism $\phi$  on $B(H)$ such that $K(H)\neq\phi(K(H))$.

Fix an orthonormal basis $\xi_j$, and consider the associated matrix units $\{e_{kj}\}$, i.e. $e_{kj}\xi=\langle \xi,\xi_j\rangle\,\xi_k$. The matrix units are characterized by $$ e_{kj}e_{st}=\delta_{js}\,e_{kt}.$$ It is easy to check that $\{\phi(e_{kj})\}$ is also a system of matrix units. If we let $\eta_j$ denote a unit vector in the range of the projector $\phi(e_{jj})$, then $\{\eta_j\}$ is an orthonormal basis (because $\sum\phi(e_{jj})=1$). 
Let $u\in B(H)$ be the unitary that sends $\xi_j\longmapsto \eta_j$. Then, for any $\xi\in H$, 
$$
\phi(e_{kj})\xi=\langle\xi,\eta_j\rangle\,\eta_k
=\langle\xi,u\xi_j\rangle\,u\eta_k=u\,\langle u^*\xi,\xi_j\rangle\,\xi_k
=ue_{kj}u^*\xi.
$$
So $\phi(e_{kj})=ue_{kj}u^*$. Now one can extend by linearity and continuity to all of $K(H)$ (and all of $B(H)$ by using wot-continuity. 
