State spaces of a von Neumann algebra and of a dense C* subalgebra.

Question

Assume that A is a von Neumann algebra on a separable Hilbert space H and D a dense separable sub-algebra of A. What type of a state on A is an extension of a state on D. Is it always a normal state or it can also be a singular state as well?

Answer 1

It can be either. For instance let $A=\ell^\infty(\mathbb N)$, and $D=c_0$. Any state on $A$ is an extension of its restriction to $D$. If you want a singular one, take any free ultrafilter $\omega$ and let $\varphi(x)=\lim_{n\to\omega} x(n)$ (or equivalently, using that $\ell^\infty(\mathbb N)=C(\beta\mathbb N)$, take $\varphi(x)=x(\omega)$, the point evaluation). You have $\varphi|_D=0$, so $\varphi$ is singular.

Less explicitly you can do the same when $A=B(H)$, $D=K(H)$. There are states which are zero on the compacts (basically, take $\varphi$ to be any state on the Calkin algebra an "lift" to $B(H)$).