David Roberts mentioned in his comments the relationship
K-theory : spin group
TMF : string groups
Let me recommend the first 6 pages of my unfinished article for a uniform construction of $SO(n)$, $Spin(n)$, and $String(n)$, which suggests the existence of a similar uniform construction of $H\mathbb R$, $KO$ (or $KU$), and $TMF$. You'll see that von Neumann algebras appear in the construction. More precisely, von Neumann algebras appear in the definition of conformal nets. The latter are functors from 1-manifolds to von Neumann algberas.
For a summary of the conjectural relationship between conformal nets and $TMF$, have a look at page 8 of this other paper of mine (joint with Chris Douglas).
Loop groups yield non-trivial examples of conformal nets. Those conformal nets are related (conjecturally) to equivariant $TMF$.
