Von Neumann's uniqueness theorem (CCR representations)

[Heidi]

Hi Pfs,
Please read this paper (equation 4):
https://ncatlab.org/nla b/files/RedeiCCRRepUniqueness.pdf
It is written: Surprise! P is a projector (has to be proved)...
where can we read the proof?

 

[Demystifier]

The correct link is
https://ncatlab.org/nlab/files/RedeiCCRRepUniqueness.pdf

 

[Heidi]

thanks Demystifier.

 

[dextercioby]

Thank you for bringing it up. I will check in von Neumann's original proof or some other source.

 

[dextercioby]

The only explicit proof is in von Neumann's original article.

 

[Heidi]

thanks, it's a good opportunity to revise my German a bit (I studied English and German when in high school)

 

[Heidi]

Tell me if this is correct:
To prove that A and AS(u,v)A only differ by a numerical factor, Von Neumann
calculates the "Kern" of A then of SA and then of ASA. As these "kerns" differ by a m
multiplicative constant k, then ASA = k A.
I would like to know how to translate the german word "Kern" in modern math english. Is it really integral kernel? or characteristic functional?
How to derive his forulas for A and SA?

 

[dextercioby]

Yes, an integral kernel is the modern term. As for the calculations themselves are all made by von Neumann. Later accounts (Putnam for example) are telegraphic, no explicit calculations

 

[Heidi]

There is also: https://mathoverflow.net/questions/24098/what-does-kernel-mean-in-integral-kernel

 

[Heidi]

In the paper Von Neumann considers three operators containing integrals. For each of them , he gives its kernel .
How to retrieve A , SA and ASA from these kernels?