Originally I had asked this question: Positive Linear Functionals on Von Neumann Algebras
I got some responses that directed me to a variety of resources, some of which I could not understand because the terminology was not introduced within the confines of what is readable on googlebooks, or even within that book, and others were inaccessible to me. (I actually own 0 of the references there, but some were possible to find within the school reading room. For the next while I will not have access to the reading room and so references will not be helpful unless they are google books readable.) I decided that perhaps it is best for me to try to solve the problem myself, and so I broke it into 2 steps.
- Every completely additive state is ultraweakly continuous. (Or equivalently weakly continuous on the unit ball.)
- Every ultraweakly continuous state is of the form described in the above link.
I actually succeeded in doing 2., and I would like to see 1 done.
It would be most helpful for the proof to be elementary, not only because I'm a beginning student in the subject of operator algebras, but because I want to see if it's possible. The statement seems simple enough. For me, elementary means it uses only introductory facts for Von Neumann algebras such as the bicommutant theorem, the Borel functional calculus, Kaplansky density, comparison of projections and other things like that. It may be possible to dig up a proof of this fact, or even an elementary one, by putting together many resources, but since MSE is partly about being a resource itself, as I understand, I hope you can keep your answer at least somewhat self-contained. For a more precise definition of self-contained, what I mean is I know, and maybe it's even true that most people who would ask this type of question would know, everything before Ch. 7 in these notes. The only problem is the proof of 7.1.7 is wrong, which is why I pursued my 2-step program above. (Dixmier can't be read from online) Many thanks in advance.