((f_x(k)F, ky), g k^{-1})\end{aligned}$$ with $f_x : G_x \to PU({\mathcal{H}})$ a fixed stable homomorphism.

Projective unitary $G$-equivariant stable bundles are the main tool to introduce twisted equivariant $K$-Theory as sections of Fredholm bundles. Other topologies have been used on the relevant operator spaces [@barcenasespinozajoachimuribe], [@barcenasvelasquezatiyahsegal]. However, they are all easily seen to agree ([@barcenasvelasquezatiyahsegal], Section 1.1. See also [@espinozauribeunitaryoperators]).

Projective unitary $G$-equivariant stable bundles have been classified in [@barcenasespinozajoachimuribe], Theorem 4.8 in page 1341. The isomorphism classes of projective unitary stable $G$-equivariant bundles over a proper $G$-CW complex $X$ are in correspondence with the third Borel Cohomology Group $H^3(X\times_G EG, {{\mathbb Z}})$. There exists an intrinsic abelian group structure on the isomorphism classes of projective unitary stable $G$-equivariant bundles for which this correspondence is an isomorphism of abelian groups.

Let $X$ be a proper $G$-CW complex. A $G$-Hilbert bundle over $X$ is a locally trivial bundle $E\to X$ with fiber on a Hilbert space ${\mathcal{H}}$ and structural group the group of unitary operators ${\mathcal{U}}({\mathcal{H}})$ with the $*$-strong operator topology.

The Bundle of Hilbert-Schmidt operators with the $*$-strong topology between Hilbert Bundles $E$ and $F$ will be denoted by $L_{HS}(E,F)$.

Let $P$ be a projective unitary stable $G$-equivariant bundle. Consider the 