Let $M$ be a von Neumann algebra and $\varphi$ be a normal weight on it, and let $A$ be its definition subalgebra. We still denote $\varphi$ the extension to $A$ as a linear positive functional. It is known that $\varphi$ is lower-ultraweakly-semicontinuous on $M^+$ (the positive elements of $M$).

Questions:

Is $\varphi$ ultraweakly continuous on $A$ (as a linear positive functional), where $A$ has the induced ultraweak topology of $M$ ? (Clearly, if $A=M$ this assertion is classical, right ?)

If we fix a Hilbert space representation of $M$ (so that $M\subseteq \mathcal{B}(H)$). Do we have that $\varphi$ is strongly or weakly continuous on A ?

(If it helps, one can suppose $\varphi$ to be a trace, semifinite and faithful).

- A third (somewhat related) question: Suppose that $B$ is a subalgebra of a von Neumann algebra, and that $f:B\to M$ is a positive linear map, such that $f$ is normal in the following sense : for any increasing net in $B^+$ with supremum in $B^+$, the image (by $f$) of this supremum is the supremum of the image of the net (the usual notion of normality, but with the hypothesis that the supremum lies in $B^+$). Do we have that $f$ is continuous (for the ultraweak topologies)?

These questions seem natural to me, but I haven't been able to locate any reference about them.