Can a unital *-homomorphism be extended to the bidual of a $\mathrm{C}^*$-algebra? Let $\mathcal{A}$ and $\mathcal{B}$ be unital $\mathrm{C}^*$-algebras, and $T:\mathcal{A}\to\mathcal{B}$ a unital *-homomorphism. Let $\mathcal{A}^{**}$ and $\mathcal{B}^{**}$ be the biduals of $\mathcal{A}$ and $\mathcal{B}$.
We have embeddings $\mathcal{A}\subset \mathcal{A}^{**}$ and $\mathcal{B}\subset \mathcal{B}^{**}$ so we can talk about $T(a)\in\mathcal{B}^{**}$ for $a\in A$, and so we have a map $\mathcal{A}\to \mathcal{B}^{**}$.

Under what kind of conditions can we extend this map to the whole of
$\mathcal{A}^{**}$ to give a map $\mathcal{A}^{**}\to
 \mathcal{B}^{**}$?


What is the correct way (i.e. topologically) to think of a sequence/net
$(a_\lambda)$ in $\mathcal{A}\subset \mathcal{A}^{**}$ converging to
an element $a\in \mathcal{A}^{**}$? Can this be done for all $a\in
 \mathcal{A}^{**}$?

In particular I am interested in the support projection $p_\varphi$ of a state $\varphi$ on $\mathcal{A}$. In general $p_\varphi\in \mathcal{A}^{**}$ is the support projection of $\omega_\varphi$, the extension of $\varphi$ to a normal state on $\mathcal{A}^{**}$ (i.e. I am interested in making sense, if possible, of $T(p_\varphi)$.
If helpful, you may assume that $\mathcal{B}=\mathcal{A}\otimes \mathcal{A}$ (minimal tensor product).
 A: Recall that for a linear map $T:X\to Y$ between normed spaces, we define its adjoint $T^*:Y^*\to X^*$ as $T^*(\phi)=\phi\circ T$. We have that $\|T^*\|=\|T\|$.
The answer is that we can always extend $*$-homomorphisms on the double duals: given a $*$-homomorphism $\varphi:A\to B$ between C*-algebras, the double adjoint map $\varphi^{**}:A^{**}\to B^{**}$ is a $*$-homomorphism on the double duals that extends $\varphi$. Moreover, $\varphi^{**}$ is ultraweakly continuous. Moreover, if $\varphi$ is injective, then so is $\varphi^{**}$.
The fact that $\varphi^{**}$ extends $\varphi$ and that $\varphi^{**}$ is ultraweakly continuous is obvious. I will show for example that $\varphi^{**}$ preserves multiplication: We have that $A$ is ultraweakly dense in $A^{**}$ by Goldstine's theorem. Let $x,y\in A^{**}$ and find norm bounded nets $(x_i),(y_i)$ in $A$ such that $x_i\to x$ and $y_i\to y$ ultraweakly. Fix an index $i_0$. Using in the 1st and last equation below the fact that multiplication is separately continuous for the ultraweak topology on bounded sets, we have that
$$\varphi^{**}(xy_{i_0})=\lim_i\varphi^{**}(x_iy_{i_0})=\lim_i\varphi(x_iy_{i_0})=\lim_i\varphi(x_i)\varphi(y_{i_0})=\varphi^{**}(x)\varphi(y_{i_0})$$
so $\varphi^{**}(xy_{i_0})=\varphi^{**}(x)\varphi^{**}(y_{i_0})$. As this is true for all indices $i_0$, taking ultraweak limits and employing the same argument yields that $\varphi^{**}$ preserves multiplication. A similar (less complicated) argument shows that $\varphi^{**}$ preserves involution.
The claim that $\varphi$ injective implies $\varphi^{**}$ injective is true more generally for linear maps between normed spaces that have closed image (an application of Hahn-Banach is needed).
With some extra work, one can also show that if $\varphi:A\to B$ is a completely positive map, then $\varphi^{**}:A^{**}\to B^{**}$ is also completely positive.
