The unitary group of any purely infinite von Neumann algebra is contractible (this is a generalization of Kuiper's theorem). 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.

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.