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 representations are just representation of $G$ and states over it are the same as functions of positive type on $G$.

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 $G$.