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Let $A$ be a von Neumann algebra, let $L^2(A)$ be the underlying Hilbert space of the standard form of $A$, and $P \subset L^2(A)$ the canonical positive cone (see for example this paper by Haagerup). We know that any normal linear positive functional $\phi$ on $A$ comes from a unique vector in $P$ and that this induces a bijection between element of $P$ and positive normal linear functional on $A$.

Is there a way to describe the addition/the scalar produt in $P$ in terms of the corresponding linear form, preferably without invoking the modular operator of Tomita's theory?

(I'm equally interested in both, and as we know that the norm corresponds to $\eta(1)$ one can go from one to the other easily...)

For example, I'm very interested in the case of the double dual algebra $C^{max}(G)^{**}$ for $G$ a discrete group: its (normal) representations are just representations of $G$ and bounded normal linear functional over it are the same as functions of positive type on $G$.

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    $\begingroup$ In your last paragraph, don't you mean that normal states on $C^*(G)^{**}$ correspond to positive-definite functions on $G$? $\endgroup$
    – Yemon Choi
    Commented Jan 13, 2016 at 17:48
  • $\begingroup$ @Yemon: indeed, and weakly continuous representations as well. $\endgroup$
    – Alain Valette
    Commented Jan 14, 2016 at 8:45
  • $\begingroup$ Sorry, this has been edited... I tend to forget to say normal or weakly continuous when talking about von Neumann algebras as it is very rare to consider something that isn't (and I shouldn't have say "states" either as I was not specifically requiring that $\eta(1)=1$). $\endgroup$
    – Simon Henry
    Commented Jan 14, 2016 at 9:08
  • $\begingroup$ @SimonHenry Singular states are occasionally important for the same reason we might consider singular measures on $[0,1]$... $\endgroup$
    – Yemon Choi
    Commented Jan 14, 2016 at 15:25

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The answer to your question is given in Definition 2.1.1, on page 34 of Kosaki's PhD thesis: https://dmitripavlov.org/scans/kosaki-thesis.pdf

I prefer however the description of $L^2(A)$ given by equation (6.1), on page 18 of my paper: http://arxiv.org/pdf/1110.5671v2.pdf

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