Given a von Neumann algebra $A$, its Haagerup $L^2$-space $H:=L^2A$ (also known as the standard form of the Neumann algebra) comes equipped with a positive cone $P\subset H$.

**Question:** How much does the pair $(H,P)$ remember about the von Neumann algebra $A$?

• Can I detect, just by looking at $(H,P)$, whether $A$ is a factor?

• How about the type of the factor?

• Can I recover $A$ up to isomorphism, just from $(H,P)$?

(I would realy like it if it were true that one can recover $A$ up to isomorphism, by just knowing at $H$ and $P$.)