Operator Theoretical Models for $K(\mathbb{Z}, 3)$

Question

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

Answer 1

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.

Answer 2

The unitary group of any purely infinite von Neumann algebra is contractible (this is a generalization of Kuiper's theorem due to Brüning and Willgerodt, “Eine Verallgemeinerung eines Satzes von N. Kuiper”). Thus the projective unitary group of any purely infinite von Neumann algebra has the homotopy type of K(Z,2) and its classifying space has the homotopy type of K(Z,3). This result has nothing to do specifically with hyperfinite type III₁ factors, they appear for a different reason in the cited paper by Stolz and Teichner.