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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

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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.

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  • $\begingroup$ Oh, nice side observation: $Out(\mathcal{O}_2 \otimes \mathbb{K})$ is what is called the Picard group of $\mathcal{O}_2$, that is the group of isomorphism classes of self-Morita equivalences. $\endgroup$ May 8, 2012 at 16:31
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    $\begingroup$ @DavidRoberts: The space $Aut(\mathcal{O}_2 \otimes \mathbb{K})$ is contractible in the pointwise norm topology. I am not quite sure, which subspace topology this induces on $PU(M(\mathcal{O}_2 \otimes \mathbb{K}))$. I think it should be the topology induced by the strict topology on $U(M(\mathcal{O}_2 \otimes \mathbb{K}))$. As mentioned in the answer I am not sure if the exact sequence of topological groups is a fibration though. $\endgroup$ Oct 20, 2017 at 8:17
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    $\begingroup$ @DavidRoberts: That is the issue. In fact, any automorphism of $\mathcal{O}_2 \otimes \mathbb{K}$ is asymptotically inner. This seems to indicate that the image of the projective unitary group should be dense. $\endgroup$ Oct 20, 2017 at 20:43
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    $\begingroup$ This makes it seem that a kind of stack quotient would be sensible... $\endgroup$
    – David Roberts
    Oct 21, 2017 at 0:40
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    $\begingroup$ @DavidRoberts: True. In particular, I think one should look at the crossed module one obtains from the homomorphism $U(M(\mathcal{O}_2 \otimes \mathbb{K})) \to Aut(\mathcal{O}_2 \otimes \mathbb{K})$. I have thought about crossed modules like this for some time, since they seem to be related to Fell bundles for example. $\endgroup$ Oct 21, 2017 at 8:32
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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.

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  • $\begingroup$ Thanks for the answer. What is the reference for the contractibility? Is there an operator theoretical model for the delooping too? $\endgroup$ May 8, 2012 at 7:23
  • $\begingroup$ Andre Henriques' answer mathoverflow.net/questions/44045/… to the question Neil mentioned points to what may be specific about hyperfinite type III factors here. $\endgroup$ May 8, 2012 at 9:13
  • $\begingroup$ @Nicolas: I added a reference for the contractibility result. $\endgroup$ May 8, 2012 at 16:19
  • $\begingroup$ @David: True, but unfortunately this relation is only conjectural. $\endgroup$ May 8, 2012 at 16:19
  • $\begingroup$ @Nicolas: The model for delooping is explained by Andrew Stacey in this answer: mathoverflow.net/questions/44045/… $\endgroup$ May 8, 2012 at 16:21

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