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 $G$.

  • 1
    $\begingroup$ In your last paragraph, don't you mean that normal states on $C^*(G)^{**}$ correspond to positive-definite functions on $G$? $\endgroup$
    – Yemon Choi
    Jan 13 '16 at 17:48
  • $\begingroup$ @Yemon: indeed, and weakly continuous representations as well. $\endgroup$ Jan 14 '16 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 $\eta(1)=1$). $\endgroup$ Jan 14 '16 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
    Jan 14 '16 at 15:25

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.