.$$

Let $\mathcal{H}$ be a Hilbert space. A continuous homomorphism $a$ defined on a Lie group $G$, $a : G \to P{\mathcal{U}}({\mathcal{H}})$ is called stable if the unitary representation ${\mathcal{H}}$ induced by the homomorphism $\widetilde{a}: \widetilde{G}= a^*{\mathcal{U}}({\mathcal{H}}) \to
{\mathcal{U}}({\mathcal{H}})$ contains each of the irreducible representations of $\widetilde{G}$

\[def projective unitary G-equivariant stable bundle\] Let $X$ be a proper $G$-CW complex. A projective unitary $G$-equivariant stable bundle over $X$ is a principal $P{\mathcal{U}}({\mathcal{H}})$-bundle $$P{\mathcal{U}}({\mathcal{H}}) \to P \to X$$ where $P{\mathcal{U}}({\mathcal{H}})$ acts on the right, endowed with a left $G$-action lifting the action on $X$ such that:

-   the left $G$-action commutes with the right $P{\mathcal{U}}({\mathcal{H}})$ action, and

-   for all $x \in X$ there exists a $G$-neighborhood $V$ of $x$ and a $G_x$-contractible slice $U$ of $x$ with $V$ equivariantly homeomorphic to $ U \times_{G_x} G$ with the action $$G_x \times (U \times G) \to U \times G, \ \ \ \
    k \cdot(u,g)= (ku, g k^{-1}),$$ together with a local trivialization $$P|_V \cong  (P{\mathcal{U}}({\mathcal{H}}) \times U) \times_{G_x} G$$ where the action of the isotropy group is: $$\begin{aligned}
     G_x \times \left( (PU({\mathcal{H}}) \times U) \times G \right)& \to & (P{\mathcal{U}}({\mathcal{H}}) \times U) \times G
     \\
    \ \left(k , ((F,y),g)\right)& \mapsto & 