TITLE: The positive cone of the standard representation of a Von Neumann algebra
QUESTION [8 upvotes]: 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 $G$.

REPLY [5 votes]: 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