von Neumann algebras and loop groups
The topologists Peter Teichner and Stephan Stolz have proposed
a geometric construction of elliptic cohomology using fusion of bimodules.
We outline several purely von Neumann algebraic questions raised
by their work. In particular, we show how the classification of stable G-kernels
in factors (due to Alain Connes for cyclic groups and Jones for finite groups)
can be extended to compact Lie groups. The characteristic invariant now lies
in Calvin Moore's Borel cohomology group H3(G,T)
=H4(BG,Z). Existence is proved using
local loop group factors; stability using the trace-class
version of BDF theory; and uniqueness using results on minimal actions
obtained with Sorin Popa.