While looking over the first chapter of

1) *Quantum Fields and Strings: A Course For Mathematicians* (P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and E. Witten, eds.,), 2 vols., American Mathematical Society, Providence, 1999.

I wondered whether there would be any use to developing a theory of super-von Neumann algebras, mimicking the usual theory. Not knowing whether or not this would be a sterile or trivial exercise, I never tried.

I have always wondered, though:

Would there be any benefit in developing a theory of super-von Neumann algebras, and if so what would the benefit likely be? Particularly, could such a theory tell us anything useful about ordinary von Neumann algebras we otherwise couldn't easily obtain?

Perhaps this is a trivial question, but I'm curious if anyone with broader knowledge can shed some light on this.

Of course, the dream is that looking at something like this would miraculously unveil something cool like a canonical time-evolution on $II_{1}$-factors.

(This is another candidate for the 'dumb question' tag!)

duhquestion. Ihaveheard Kawahigashi mention super von Neumann algebras in connection with is work (w/ Roberto Longo) on QFTs, but I don't know what's going on enough to ask a truly intelligent question about it! – Jon Bannon Oct 26 '10 at 15:10