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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 $P$.)

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    $\begingroup$ 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. $\endgroup$
    – MaoWao
    Commented Dec 9, 2023 at 19:35

1 Answer 1

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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.

The article by Connes works with cyclic and separating vectors, so I don't see immediately if this result generalizes beyond the countably decomposable case as expected.

[1] Connes. Caractérisation des espaces vectoriels ordonnés sous jacents aux algèbres de Von Neumann, 1974.

[2] Kadison. On isometries of operator algebras, 1951.

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