Hilbert Grassmannians as classifying spaces

TL;DR

This work shows that for a separable infinite‑dimensional Hilbert space, the finite‑rank unitary classifying spaces $B\mathbb{U}(n)$ are modeled by both the uniform and weak Hilbert Grassmannians ${^{\rm u}\mathscr{G}_n(\mathfrak{h})}$ and ${^{\rm w}\mathscr{G}_n(\mathfrak{h})}$ via homotopy equivalences. It develops a Serre‑fibration framework to compute homotopy groups and proves fixed‑rank identifications, while revealing no‑go results for the infinite case in both the uniform and weak settings, except under the fixed‑rank perspective. The results connect these models to vector bundle classification and reduced K‑theory, showing that $[X,{^{\rm w}\mathscr{G}_n(\mathfrak{h})}]\simeq Vec_\mathbb{C}^n(X) \simeq \widetilde{K}^0(X)$ for suitable $X$, and establish a continuity property for states induced by projections that underscores the physical relevance of the Hilbert Grassmannian approach.

Abstract

In this short work we prove that the Hilbert Grassmannians endowed with the weak topology are models for the classifying spaces of the unitary groups. As application of this result one can use Hilbert Grassmannians for the presentation of the $K$-theory of topological spaces by computing equivalences classes of homotopy equivalent maps.

Theorems & Definitions (18)

• Theorem 1.1: Hilbertian classifying spaces
• Corollary 1.2
• Remark 1.3: Chern classes
• Remark 1.4: Reduced $K$-theory
• Proposition 1.5
• Proposition 3.1
• Corollary 3.2
• Proposition 3.3
• Corollary 3.4
• Lemma 4.1