Two possible ideas. One is that you can realize $K(\mathbb{Z},3)$ as the quotient $U(HS)/PU(\infty)$ where $U(HS)$ is the unitary group on the Hilbert space of Hilbert-Schmidt operators. However, I don’t know if you can see the group structure in this model to make a principal bundle. This construction is in https://arxiv.org/pdf/hep-th/9702147.pdf.

There is also Andre Henriques’s conjecture that the outer automorphisms of a hyperfinite type III factor model $K(\mathbb{Z},3)$. See ‘Naturally occurring’ $K(\pi, n)$ spaces, for $n \geq 2$

Two possible ideas. One is that you can realize $K(\mathbb{Z},3)$ as the quotient $U(HS)/PU(\infty)$ where $U(HS)$ is the unitary group on the Hilbert space of Hilbert-Schmidt operators. However, I don’t know if you can see the group structure in this model to make a principal bundle. This construction is in

• Alan L. Carey, Diarmuid Crowley, Michael K. Murray, Principal Bundles and the Dixmier Douady Class, Commun.Math.Phys. 193 (1998) pp 171-196, doi:10.1007/s002200050323, arXiv:hep-th/9702147.

There is also Andre Henriques’s conjecture that the outer automorphisms of a hyperfinite type III factor model $K(\mathbb{Z},3)$. See ‘Naturally occurring’ $K(\pi, n)$ spaces, for $n \geq 2$