Non normal state

Question

Does anybody know an example of a state on a von Neumann algebra that is not normal? If it has relevance to physics it would be nice.

Answer 1

On $B(H)$, any state that is zero on the compacts will be non-normal. That would be any state that comes from a state on the Calkin algebra.

In the same spirit, take $M=\ell^\infty(\mathbb N)$. It is well-known that $M=C(\beta\mathbb N)$, where $\beta\mathbb N$ is the Stone-Cech compactification of $\mathbb N$. The extreme states of $M$ are the point evaluations; any point evaluation at $t\in\beta\mathbb N\setminus\mathbb N$ (that is, a limit along a free ultrafilter) will be a state that is zero on "the compacts" (i.e., $c_0$), and thus will not be normal (as the identity is the sot-limit of the characteristics of finite sets).

In general, I wouldn't expect non-normal states to be very explicit. Because if you can write a "formula" for your state it is likely to be nice, in the same way that you cannot find many pathologies in continuity and differentiability by writing formulas (since the natural operations are "good").