David Roberts mentioned in his comments the relationship

<b>K-theory : spin group<br>
TMF : string groups</b>

Let me recommend the first 6 pages of my  [unifinished article][1]
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 appears in the construction.
More precisely, von Neumann algebras appear in the definition of <i>conformal nets</i>. 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][2] of mine.

Loop groups yield non-trivial examples of conformal nets.
Those conformal nets are related (conjecturally) to equivariant $TMF$.


  [1]: http://www.staff.science.uu.nl/~henri105/PDF/TringWP.pdf
  [2]: http://arxiv.org/pdf/1103.4187v1.pdf