# Positive cone in Haagerup L²-space: how much information does it contain?

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 $$P$$.)

• If $A$ is finite, the map $a\mapsto a^{\mathrm{op}}$ extends to a unitary from $L^2(A)$ to $L^2(A^{\mathrm{op}})$ that sends $L^2_+(A)$ onto $L^2_+(A^{\mathrm{op}})$. However, there are $\mathrm{II}_1$ factors that are not isomorphic to their dual. Dec 9, 2023 at 19:35

For countably decomposable von Neumann algebras, the pair $$(L^2(M),L^2_+(M))$$ characterizes $$M$$ up to Jordan $$\ast$$-isomorphism by the result of Section 3 from [1]. Moreover, a Jordan $$\ast$$-isomorphism between von Neumann algebras is the direct sum of a $$\ast$$-isomorphism and a $$\ast$$-anti-isomorphism by Theorem 10 in [2].
In particular, one can detect factoriality and the type of $$M$$ from $$(L^2(M),L^2_+(M))$$, and in the case of factors, any two von Neumann algebras $$M$$, $$N$$ with $$(L^2(M),L^2_+(M))\cong (L^2(N),L^2_+(N))$$ are $$\ast$$-isomorphic or $$\ast$$-anti-isomorphic.