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.