Let $varphicolon Ato B$ be a bounded, linear map between C*-algebras. Is the bitranspose $varphi^{**}colon A^{**}to B^{**}$ continuous when the von Neumann algebras $A^{**}$ and $B^{**}$ are equipped with their $sigma$-strong topologies?

Motivation/Background: Note that $varphi^{**}$ is clearly continuous when $A^{**}$ and $B^{**}$ are equipped with their $sigma$-weak topologies, since these agree with the weak${}^*$-topology from the preduals, and $varphi^{**}$ is weak${}^*$-continuous (that is, $sigma(A^{**},A^*)-sigma(B^{**},B^*)$-continuous).

If $varphi$ if completely positive, then it follows that $varphi^{**}$ is a completely positive, normal map, and therefore is continuous for the $sigma$-strong topologies. Thus, the question is only interesting if $varphi$ is not completely positive.

On bounded sets of a von Neumann algebra, the $sigma$-strong topology agrees with the strong (operator) topology (SOT). If $Msubseteq B(H)$ is a von Neumann algebra, then a net $(a_j)_j$ in $M$ SOT-converges to $ain M$ if $|a_jxi-axi|to 0$ for every $xiin H$.