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. 



Continuity of a weight on its definition domain in a von Neumann algebra
1) if $A=M$ the assertion is indeed classical: normal states are exactly those ultraweakly continuous. But consider the case where $M=B(H)$ and $\varphi$ is the trace. Then the definition subalgebra $A$ is exactly the trace-class operators. Let $\{p\_k\}\subset A$ be a maximal net of pairwise orthogonal projections of trace 1 (i.e. $\{e\_{kk}\}$ for any choice of matrix units). Then $p\_k\to0$ ultraweakly, but $\mbox{Tr}(p\_k)=1$ for all $k$. So the trace is not ultraweakly continuous on the definition algebra, only ultraweakly lower-semicontinuous. 


2) The example on 1) is already explicitly represented, so no.


3) Still thinking about it.


