I am looking for a reference concerning operator theoretical Models of $K(\mathbb{Z},3)$. Stolz-Teichner briefly say in "what is an elliptic object" that a certain hyperfinite Type III-factor, called "local fermions on the circle" should be the right thing. Thanks 



Operator Theoretical Models for $K(\mathbb{Z}, 3)$
Here is a $C^\*$-algebraic version of the model described in Andre Henriques' answer (the latter was linked by David Corfield in the accepted answer above): 


Let $\mathcal{O}\_2$ be the Cuntz algebra generated by two partial isometries $s\_1$ and $s\_2$ subject to the relations $s\_i^\*s\_j = \delta\_{i,j}$ and $s\_1s\_1^\* + s\_2s\_2^\* = 1$. This algebra has vanishing $K$-theory as was calculated by Cuntz. By the universal coefficient theorem and Bott periodicity, $KK(\mathcal{O}\_2, S^n\mathcal{O}\_2)$ should vanish as well, where $S^n\mathcal{O}\_2$ denotes the $n$-fold suspension of $\mathcal{O}\_2$. The automorphism group of the stabilized algebra $\mathcal{O}\_2 \otimes \mathbb{K}$ (where $\mathbb{K}$ denote the compact operators on a separable Hilbert space) fits into a short-ish exact sequence


$$1 \to U(1) \to U(M(\mathcal{O}\_2 \otimes \mathbb{K})) \to Aut(\mathcal{O}\_2 \otimes \mathbb{K}) \to Out(\mathcal{O}\_2 \otimes \mathbb{K}) \to 1$$ 


where $M(\mathcal{O}\_2 \otimes \mathbb{K})$ is the multiplier algebra. The homotopy groups of $Aut(A \otimes \mathbb{K})$ for so-called Kirchberg algebras have been calculated (yeah, I was surprised too :-). You can find them in a paper by Dadarlat called "The homotopy groups of the automorphism groups of Kirchberg algebras". The result is 


$$\pi\_n(Aut(A \otimes \mathbb{K})) \cong KK(A,S^nA).$$


Now, $\mathcal{O}\_2$ fits into that class and therefore has weakly contractible automorphism groups, but - by a theorem of Mingo - $U(M(\mathcal{O}\_2 \otimes \mathbb{K}))$ is contractible as well. Analyzing the above sequence, we see that $Out(\mathcal{O}\_2 \otimes \mathbb{K})$ has the weak homotopy type of a $K(\mathbb{Z},3)$... at least if


$$1 \to PU(M(\mathcal{O}\_2 \otimes \mathbb{K})) \to Aut(\mathcal{O}\_2 \otimes \mathbb{K}) \to Out(\mathcal{O}\_2 \otimes \mathbb{K}) \to 1$$ 


is a fibration. In fact, it could very well be that the topology on the quotient $Out(\mathcal{O}\_2 \otimes \mathbb{K})$ is quite horrible.


