Norm functionals of $B(H)$ restricted to sub von-Neumann algebras Let $H$ be a Hilbert space, we know that weak topology over $B(H)$, operator algebra of bounded linear operators from $H$ into $H$, is the topology generated by 
$\{\langle \cdot \xi,\eta\rangle:\; \xi,\eta\in H\}$. 
So naturally, I think about the norm of $\langle \cdot \xi,\eta\rangle$ as a linear functional over $V$ a von Neumann subalgebra of $B(H)$. And I guess that
$$\|\langle\cdot \xi,\eta\rangle\|=\inf \{\|\xi'\|_H \|\eta'\|_H:\; s.t.
\;\langle T \xi',\eta'\rangle = \langle T \xi,\eta\rangle\; \forall T\in V\}.$$
But I am not sure how can I show that. Indeed I am wondering whether this is correct or not even!
 A: The answer is "yes".  The following ideas can be found in standard texts.
The functional $\omega:T\mapsto \langle T\xi,\eta\rangle$ is normal, so it has a polar decomposition: $\omega = v |\omega|$ where $v\in V$ is a partial isometry, and $|\omega|$ is positive.  $v$ is uniquely defined by the extra condition that $v^*v=s(|\omega|)$ the support projection of $|\omega|$.  So $v^*\omega = v^*v|\omega| = |\omega|$.  However, obviously $v^*\omega=\langle\cdot v^*\xi,\eta\rangle$.
At this point I cheat, and invoke a result from Kadison+Ringrose, Vol II, Prop 7.3.12.  This says that as $|\omega|=\langle\cdot v^*\xi,\eta\rangle$ is positive, we can find $\xi'\in H$ with $|\omega|=\langle\cdot \xi',\xi'\rangle$.  Clearly $\| |\omega| \| = \|\xi'\|^2$.  As $e=v^*v$ is the support projection of $|\omega|$ it is central, and so we see that $|\omega| = \langle \cdot \xi',\xi'\rangle
= \langle \cdot e\xi',e\xi'\rangle$.  It follows that $\| |\omega| \| = \|\xi'\|^2
= \|e\xi'\|^2$ and so $e\xi'=\xi'$.  So also $\|v\xi'\|=\|\xi'\|$.
Now, $\omega = \langle \cdot v\xi',\xi'\rangle$, and so  $\|\omega\|
= \||\omega|\| = \|\xi'\|^2 = \|v\xi'\| \|\xi'\|$.
So your formula for the norm is correct, and actually the infimum is obtained.
I will admit that this was harder than I thought.  I wonder if anyone else has an
easier proof?
Edit: A general comment.  Very often, we work with von Neumann algebras in "standard position".  From your
bio I see you are interested in abstract harmonic analysis, so perhaps interested
in group von Neumann algebras $VN(G)$.  Acting on $L^2(G)$ these are in standard
position.  Then every normal functional arises as $\langle \cdot \xi,\eta \rangle$
for some $\xi,\eta$.  So in this case, I wouldn't need to use the reference, and
the argument becomes a lot easier.
